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On a number line, how many numbers locate a point, and what is the number called?	One. It is called the coordinate of the point.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
On a number line, what is the point that corresponds to a number called?	The graph of the number. The number is the coordinate of the point.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
Why say coordinate rather than x-axis number?	Coordinate works for any axis, any dimension and any system, including polar, whose numbers lie on no axis. And a number on the x-axis is a point (3, 0), while the x-coordinate of (3, 5) is read off the axis without lying on it.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
Is the x-coordinate of (3, 5) the point (3, 0) on the x-axis?	No. (3, 0) is the foot of the perpendicular from (3, 5) to the x-axis, a point with two coordinates. The x-coordinate is the number 3 the axis reads there, and it is what equations compute with.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Does every coordinate lie on an axis?	No. Rectangular coordinates do, one axis each; polar coordinates are a distance and an angle. A coordinate is one slot in the tuple that fixes a position, and the axis label only says which slot.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Is abscissa used for a point on a plain number line?	No. Abscissa is the x-coordinate of a point in the plane. On a line the word is coordinate.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
In the plane, how many numbers locate a point, and what are they called together?	Two. Together they are the ordered pair (x, y), the coordinates of the point.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Is (x, y) one coordinate or two?	Two: x is the x-coordinate and y is the y-coordinate. The pair is the point's coordinates, plural. Everyday speech says a coordinate for a whole location; the textbooks do not.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What is the x-axis?	The horizontal number line.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What is the y-axis?	The vertical number line.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
In the xy-plane, what is the origin O?	The point where the x-axis and the y-axis intersect. Its coordinates are (0, 0).	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What does the number of axes of a coordinate system tell you?	The dimension of the space: one axis per dimension and one coordinate per axis. A line has one, the plane two, space three.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What are the coordinates of the origin on a line, in the plane, and in space?	0 on a line, (0, 0) in the plane, (0, 0, 0) in space: one 0 per axis.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Is the origin a special point of the plane?	No. It is whichever point was chosen as 0 on both axes when they were drawn. Its coordinates are (0, 0) by definition.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Must the x-axis and the y-axis use the same scale?	No. Mathematics usually does; an application may use a different scale on each axis.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Which points on the x-axis are positive, and which negative?	Right of O positive, left of O negative.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Which points on the y-axis are positive, and which negative?	Above O positive, below O negative.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What does the arrowhead at the end of an axis mean?	It marks the positive direction of that axis.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Can an axis have arrowheads on both ends?	Yes. Then they mean the line goes on forever, and the numbers along the axis say which way is positive.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
Why does it matter which direction of an axis is positive?	It sets the orientation: with x right and y up, the turn from the x-axis to the y-axis is counterclockwise. Flip one axis and every rotation and slope changes sign on the picture.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
A computer screen has y positive downward. What changes?	The numbers do not; the picture does. Counterclockwise becomes clockwise, so a textbook rotation formula turns the wrong way.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
What are the two names of the coordinate system built from two perpendicular number lines?	Rectangular coordinate system, or Cartesian coordinate system, after René Descartes.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What is the xy-plane?	The plane in which the x-axis and the y-axis lie.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What other names does the xy-plane go by?	The Cartesian plane, or the coordinate plane.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
Why does Axler say the coordinates are called rectangular?	Each point's coordinates are determined by a rectangle: corners at the origin, the point, and its two feet on the axes, with sides |x| and |y|.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What names does Axler prefer to x-axis, y-axis, x-coordinate and y-coordinate?	Horizontal axis and vertical axis; first coordinate and second coordinate. They name the slot, so they survive an axis called t and a point written (y, x).	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What two choices fix the scale on an axis?	Where 0 goes, the origin, and where 1 goes, the unit point. Every other label is then determined.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
What is genuinely different from rectangular coordinates?	Polar coordinates: the same point named by its distance from the origin and its angle from the x-axis.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What are the coordinate axes?	The x-axis and the y-axis.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
In the ordered pair (x, y) of a point, what does x measure?	The signed distance of the point from the y-axis: how far right or left.	math precalculus rectangular-coordinates signed-distance	Math::Rectangular coordinates
In the ordered pair (x, y) of a point, what does y measure?	The signed distance of the point from the x-axis: how far up or down.	math precalculus rectangular-coordinates signed-distance	Math::Rectangular coordinates
What does signed mean in signed distance?	The sign says which side: x > 0 right of the y-axis, x < 0 left of it.	math precalculus rectangular-coordinates signed-distance	Math::Rectangular coordinates
The x-coordinate measures distance from which axis?	The y-axis. Right or left is measured from the vertical line.	math precalculus rectangular-coordinates signed-distance	Math::Rectangular coordinates
What does Larson call the signed distance of a point from an axis?	The directed distance. Two names, one number.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What is the ordered pair (x, y) of a point P called?	The coordinates of P.	math precalculus rectangular-coordinates definition	Math::Rectangular coordinates
How do you plot the point (−3, 1)?	Go 3 units left of O along the x-axis, then 1 unit straight up, and place a dot.	math precalculus rectangular-coordinates plotting	Math::Rectangular coordinates
What does it mean to plot a point?	To place a dot at the location its coordinates give.	math precalculus rectangular-coordinates plotting	Math::Rectangular coordinates
Must the coordinates of a point be whole numbers?	No. (−1.5, −2.5) is a point in quadrant III. The grid lines are only a reading aid.	math precalculus rectangular-coordinates plotting	Math::Rectangular coordinates
What form do the coordinates of a point on the x-axis have?	(x, 0): the y-coordinate is 0.	math precalculus rectangular-coordinates axes	Math::Rectangular coordinates
What form do the coordinates of a point on the y-axis have?	(0, y): the x-coordinate is 0.	math precalculus rectangular-coordinates axes	Math::Rectangular coordinates
A point has y-coordinate 0. Where is it?	On the x-axis.	math precalculus rectangular-coordinates axes	Math::Rectangular coordinates
A point has x-coordinate 0. Where is it?	On the y-axis.	math precalculus rectangular-coordinates axes	Math::Rectangular coordinates
What is another name for the x-coordinate of a point?	The abscissa.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What is another name for the y-coordinate of a point?	The ordinate.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What is the abscissa of a point?	Its x-coordinate.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What is the ordinate of a point?	Its y-coordinate.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
Where do the words abscissa and ordinate come from?	Latin, 1600s: abscissa means cut off, ordinata means applied in order. Older than the letters x and y.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
Is the abscissa the x-axis?	No. The abscissa is the number x. The line is the x-axis, or the axis of abscissas.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What is the third coordinate of a point in space called, when it has a name?	The applicate, rare in English; kota in Polish, cote in French. Usually just the z-coordinate.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What are abscissa and ordinate called in Polish, French and German?	Odcięta and rzędna; abscisse and ordonnée; Abszisse and Ordinate.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What is the Polish word for coordinate?	Współrzędna, plural współrzędne: współ- is co- and rzędna is ordinate, the Latin word rebuilt from Polish parts. A coordinate system is układ współrzędnych.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
What are quadrant, origin and ordered pair in Polish?	Ćwiartka (pierwsza ćwiartka is quadrant I), początek układu współrzędnych, para uporządkowana.	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
How is the point P whose coordinates are (x, y) usually written and said?	P = (x, y), and simply the point (x, y).	math precalculus rectangular-coordinates vocabulary	Math::Rectangular coordinates
The notation (x, y) has three meanings. What are they?	A point in the plane; the open interval of all t with x < t < y; a vector, x right and y up.	math precalculus rectangular-coordinates notation	Math::Rectangular coordinates
Can (5, 2) be an open interval?	No. An interval (a, b) needs a < b. (5, 2) must be the point.	math precalculus rectangular-coordinates notation	Math::Rectangular coordinates
Does a vector start at the origin?	Not necessarily. A vector is a displacement and can start anywhere. Started at O, the vector (3, 1) ends at the point (3, 1).	math precalculus rectangular-coordinates notation	Math::Rectangular coordinates
What is the vector from P = (2, 3) to Q = (5, 4)?	(5 − 2, 4 − 3) = (3, 1): the coordinates of Q minus those of P.	math precalculus rectangular-coordinates notation	Math::Rectangular coordinates
What are the quadrants?	The four regions the axes divide the plane into, numbered I to IV counterclockwise from the top right.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Signs of x and y in quadrant I?	x > 0 and y > 0 (top right).	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Signs of x and y in quadrant II?	x < 0 and y > 0 (top left).	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Signs of x and y in quadrant III?	x < 0 and y < 0 (bottom left).	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Signs of x and y in quadrant IV?	x > 0 and y < 0 (bottom right).	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
A point has x < 0 and y > 0. Which quadrant?	Quadrant II.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
A point has x > 0 and y < 0. Which quadrant?	Quadrant IV.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
A point has x < 0 and y < 0. Which quadrant?	Quadrant III.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
A point has x > 0 and y > 0. Which quadrant?	Quadrant I.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Which quadrant is a point on a coordinate axis in?	None. 0 has no sign, and a quadrant is a pair of signs.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Which quadrant is (3, −2) in?	Quadrant IV: x > 0, y < 0.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Which quadrant is (−2, −3) in?	Quadrant III: x < 0, y < 0.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Which quadrant is (−3, 1) in?	Quadrant II: x < 0, y > 0.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Which quadrant is (3, 2) in?	Quadrant I: x > 0, y > 0.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
In which order are the quadrants numbered around the origin?	Counterclockwise: I top right, II top left, III bottom left, IV bottom right.	math precalculus rectangular-coordinates quadrants	Math::Rectangular coordinates
Why does an angle open from the x-axis toward the y-axis, and not the other way?	By definition: a positive turn carries the first axis onto the second. So a small positive angle lands at a point with positive y, and sin θ is positive just above 0°. Clockwise is only a different convention, as on a clock or a compass.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
Why are the quadrants numbered counterclockwise?	An angle opens from the positive x-axis toward the positive y-axis, which with x right and y up is counterclockwise. Its first quarter turn, 0° to 90°, sweeps quadrant I, and II, III, IV follow the turn.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
Are (2, 5) and (5, 2) the same point?	No. (2, 5) is 2 right and 5 up; (5, 2) is 5 right and 2 up.	math precalculus rectangular-coordinates ordered-pair	Math::Rectangular coordinates
Why is a quadrant nothing but a pair of signs?	Membership depends on the signs of x and y, never on their sizes: (1, 1) and (1000, 5) share quadrant I.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
What happens to the quadrant of (x, y) when x is replaced by −x?	The point reflects across the y-axis: I and II swap, III and IV swap.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
In which quadrants is the product xy positive?	I and III, where x and y have the same sign. Negative in II and IV.	math precalculus rectangular-coordinates reasoning	Math::Rectangular coordinates
State the distance formula.	d(P1, P2) = the square root of (x2 - x1)^2 + (y2 - y1)^2.	math precalculus distance-formula formula	Math::Distance formula
The distance formula in words?	Subtract the x's and square, subtract the y's and square, add, take the square root.	math precalculus distance-formula formula	Math::Distance formula
Which theorem is the distance formula?	The Pythagorean theorem, with the legs read off the coordinates.	math precalculus distance-formula why	Math::Distance formula
Distance formula: where is the right angle?	At P3 = (x2, y1), where a horizontal line through P1 meets a vertical line through P2.	math precalculus distance-formula why	Math::Distance formula
Distance formula: what are the two legs?	The horizontal leg |x2 - x1| and the vertical leg |y2 - y1|.	math precalculus distance-formula why	Math::Distance formula
Why does the order of the two points not matter?	Swapping them flips the sign of each difference, and squaring erases the sign.	math precalculus distance-formula why	Math::Distance formula
Why does the distance formula need no absolute values?	The legs are |x2 - x1| and |y2 - y1|, but |t|^2 = t^2, so the squares do the job.	math precalculus distance-formula why	Math::Distance formula
What is the square root of t^2?	|t|, not t. The square root is never negative.	math precalculus distance-formula why	Math::Distance formula
Horizontal points (x1, y) and (x2, y): what does the formula give?	|x2 - x1|, the distance on a number line.	math precalculus distance-formula why	Math::Distance formula
How do you compare two distances without taking square roots?	Compare their squares: for non-negative numbers, squaring keeps the order.	math precalculus distance-formula why	Math::Distance formula
When is d(P1, P2) = 0?	Only when P1 = P2: a sum of two squares is 0 only when both are 0.	math precalculus distance-formula why	Math::Distance formula
Distance from (0, 0) to (a, b)?	The square root of a^2 + b^2.	math precalculus distance-formula formula	Math::Distance formula
Would the distance formula be right on axes that are not perpendicular?	No. It needs the right angle between horizontal and vertical.	math precalculus distance-formula why	Math::Distance formula
Distance from (1, 3) to (5, 6)?	5. Legs 4 and 3, corner at (5, 3).	math precalculus distance-formula problems	Math::Distance formula
Distance from (-4, 5) to (3, 2)?	The square root of 58, about 7.62: 49 + 9.	math precalculus distance-formula problems	Math::Distance formula
Distance from (0, 0) to (1, 1)?	The square root of 2, about 1.414.	math precalculus distance-formula problems	Math::Distance formula
Distance from (-2, 4) to (5, 4)?	7. Same y, so |5 - (-2)|.	math precalculus distance-formula problems	Math::Distance formula
Simplify the square root of 20.	2 times the square root of 5, since 20 = 4 times 5.	math precalculus distance-formula problems	Math::Distance formula
A = (-2, 1), B = (2, 3), C = (3, 1): right triangle?	Yes. AB^2 = 20, BC^2 = 5, AC^2 = 25 = 20 + 5. The right angle is at B.	math precalculus distance-formula problems	Math::Distance formula
Closer to the origin: (5, -1) or (-3, 4)?	(-3, 4): 25 against 26, comparing squares.	math precalculus distance-formula problems	Math::Distance formula
Points on the x-axis at distance 5 from (1, 3)?	(5, 0) and (-3, 0): (x - 1)^2 + 9 = 25.	math precalculus distance-formula problems	Math::Distance formula
How can distances show three points lie on one line?	The two short distances add up exactly to the long one.	math precalculus distance-formula why	Math::Distance formula
What is the triangle inequality in coordinates?	d(P, R) is at most d(P, Q) + d(Q, R), equal only when Q is on the segment PR.	math precalculus distance-formula why	Math::Distance formula
Trap: the square root of a^2 + b^2 equals a + b?	No. The square root of 9 + 16 is 5, not 7.	math precalculus distance-formula trap	Math::Distance formula
Trap: (x2 - x1)^2 + (y2 - y1)^2 equals (x2 - x1 + y2 - y1)^2?	No. From (0, 0) to (3, 4): 9 + 16 = 25, but 7^2 = 49.	math precalculus distance-formula trap	Math::Distance formula
Trap: from (3, -1) to (3, -7) the distance is -6?	No. A distance is never negative: 6.	math precalculus distance-formula trap	Math::Distance formula
Trap: (3 - -4)^2 = (3 - 4)^2 = 1?	No. 3 - (-4) = 7, so the square is 49. Put negative coordinates in brackets.	math precalculus distance-formula trap	Math::Distance formula
Trap: the subscript in x2 means x squared?	No. x2 is the x-coordinate of the second point; a square is written x^2.	math precalculus distance-formula trap	Math::Distance formula
Trap: -3^2 = 9?	No. -3^2 = -9; the square applies before the minus. (-3)^2 = 9.	math precalculus distance-formula trap	Math::Distance formula
Trap: to compare two distances you must compute both square roots?	No. Compare the squares: 26 against 25 already decides it.	math precalculus distance-formula trap	Math::Distance formula
State the midpoint formula.	M = ((x1 + x2)/2, (y1 + y2)/2): the average of the x's and the average of the y's.	math precalculus midpoint-formula formula	Math::Midpoint formula
The midpoint formula in three words?	Sum over two, for each coordinate.	math precalculus midpoint-formula formula	Math::Midpoint formula
Each coordinate of the midpoint is what kind of average?	The arithmetic mean of the two coordinates.	math precalculus midpoint-formula formula	Math::Midpoint formula
On a number line, the point halfway between a and b?	(a + b)/2.	math precalculus midpoint-formula formula	Math::Midpoint formula
Halfway along a segment means which two things?	Equally far from both ends, and on the segment.	math precalculus midpoint-formula why	Math::Midpoint formula
Is every point equally far from A and B the midpoint of AB?	No. The whole perpendicular bisector is equidistant; only one of its points is on the segment.	math precalculus midpoint-formula why	Math::Midpoint formula
Why is halfway across and halfway up also halfway along the diagonal?	The triangle from P1 to M is the triangle from P1 to P2 at half the size: similar, scale factor 1/2.	math precalculus midpoint-formula why	Math::Midpoint formula
How far is the midpoint from each end?	Half the length of the segment.	math precalculus midpoint-formula why	Math::Midpoint formula
Does the midpoint depend on which point is called P1?	No: x1 + x2 = x2 + x1.	math precalculus midpoint-formula why	Math::Midpoint formula
The point a fraction t of the way from P1 to P2?	P1 + t(P2 - P1). t = 1/2 is the midpoint.	math precalculus midpoint-formula formula	Math::Midpoint formula
Which values of t in P1 + t(P2 - P1) give the segment?	t from 0 to 1. Outside that range the point is on the line but past an end.	math precalculus midpoint-formula why	Math::Midpoint formula
Given one end P1 and the midpoint M, the other end?	P2 = 2M - P1, coordinate by coordinate.	math precalculus midpoint-formula formula	Math::Midpoint formula
Midpoint of (-5, 5) and (3, 1)?	(-1, 3).	math precalculus midpoint-formula problems	Math::Midpoint formula
Midpoint of (1.5, -2) and (-3.5, 4)?	(-1, 1).	math precalculus midpoint-formula problems	Math::Midpoint formula
One end (1, 2), midpoint (4, -1): other end?	(7, -4).	math precalculus midpoint-formula problems	Math::Midpoint formula
One end (0, 0), midpoint (3, 5): other end?	(6, 10).	math precalculus midpoint-formula problems	Math::Midpoint formula
The center of a circle from the two ends of a diameter?	The midpoint of the diameter.	math precalculus midpoint-formula problems	Math::Midpoint formula
How can midpoints test for a parallelogram?	Its diagonals bisect each other: they have the same midpoint.	math precalculus midpoint-formula problems	Math::Midpoint formula
The average of three points, and its name for a triangle?	The centroid, ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).	math precalculus midpoint-formula problems	Math::Midpoint formula
Trap: the midpoint of (2, 3) and (10, 7) is ((10 - 2)/2, (7 - 3)/2) = (4, 2)?	No. That is half the trip. The midpoint is the sum over two: (6, 5).	math precalculus midpoint-formula trap	Math::Midpoint formula
Trap: one end (1, 2) and midpoint (4, -1), so the other end is the average ((1 + 4)/2, (2 - 1)/2)?	No. That is a quarter of the way. The other end is 2M - P1 = (7, -4).	math precalculus midpoint-formula trap	Math::Midpoint formula
Trap: Q is 5 from A and 5 from B, so Q is the midpoint of AB?	Not unless Q is on the segment, which needs AB = 10.	math precalculus midpoint-formula trap	Math::Midpoint formula
Trap: midpoint of (-3, 4) and (3, -4) is undefined because the sums are 0?	No. It is (0, 0), the origin. A zero coordinate is a perfectly good coordinate.	math precalculus midpoint-formula trap	Math::Midpoint formula
Trap: which formulas use sums and which use differences?	Midpoint: sums. Distance and slope: differences.	math precalculus midpoint-formula trap	Math::Midpoint formula
What is the graph of an equation in x and y?	The set of all points (x, y) whose coordinates satisfy the equation.	math precalculus graphs intercepts symmetry definition	Math::Graphs, intercepts and symmetry
How do you tell whether a point is on a graph?	Substitute its coordinates. On the graph exactly when the equation holds.	math precalculus graphs intercepts symmetry definition	Math::Graphs, intercepts and symmetry
Is (1, -3) on y = x^2 - 4?	Yes: 1 - 4 = -3.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Is (0, 4) on y = x^2 - 4?	No: 0 - 4 = -4. The point (0, -4) is.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
How are the graph of an equation and its solution set related?	They are the same set: each solution (x, y) is a point of the graph.	math precalculus graphs intercepts symmetry definition	Math::Graphs, intercepts and symmetry
How do you find the x-intercepts?	Set y = 0 and solve for x.	math precalculus graphs intercepts symmetry intercepts	Math::Graphs, intercepts and symmetry
How do you find the y-intercepts?	Set x = 0 and solve for y.	math precalculus graphs intercepts symmetry intercepts	Math::Graphs, intercepts and symmetry
Why set y = 0 for the x-intercepts?	The x-intercepts are on the x-axis, and every point of the x-axis has y = 0.	math precalculus graphs intercepts symmetry intercepts	Math::Graphs, intercepts and symmetry
In Sullivan, is an intercept a number or a point?	A number: the coordinate. The x-intercept 3 means the point (3, 0).	math precalculus graphs intercepts symmetry intercepts	Math::Graphs, intercepts and symmetry
Intercepts of y = x^2 - 4?	x-intercepts -2 and 2; y-intercept -4.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Intercepts of 4x^2 + 9y^2 = 36?	x-intercepts -3 and 3; y-intercepts -2 and 2.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
x-intercepts of y = x^2 + 1?	None: x^2 = -1 has no real solution.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Intercepts of y = x^3 - x?	x-intercepts -1, 0, 1; y-intercept 0.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Test for symmetry about the y-axis?	Replace x by -x. An equivalent equation means symmetric.	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Test for symmetry about the x-axis?	Replace y by -y. An equivalent equation means symmetric.	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Test for symmetry about the origin?	Replace x by -x and y by -y. An equivalent equation means symmetric.	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Symmetric about the y-axis and (3, -2) is on the graph: which other point?	(-3, -2).	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Symmetric about the origin and (3, -2) is on the graph: which other point?	(-3, 2).	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Symmetric about the x-axis and (3, -2) is on the graph: which other point?	(3, 2).	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Symmetries of y = x^2 - 4?	y-axis only.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Symmetries of x = y^2?	x-axis only.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Symmetries of y = x^3?	Origin only.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
Symmetries of x^2 + y^2 = 25?	All three: x-axis, y-axis, origin.	math precalculus graphs intercepts symmetry problems	Math::Graphs, intercepts and symmetry
A graph symmetric about both axes: what else?	Symmetric about the origin: reflecting in both axes is a half turn.	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
How do you prove a graph is NOT symmetric?	Find one point on it whose mirror image is not on it.	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
What are y-axis and origin symmetry called for functions?	Even and odd functions.	math precalculus graphs intercepts symmetry symmetry	Math::Graphs, intercepts and symmetry
Trap: for x-intercepts, set x = 0?	No. Set y = 0: the x-intercepts lie on the x-axis, where y = 0.	math precalculus graphs intercepts symmetry trap	Math::Graphs, intercepts and symmetry
Trap: symmetric about the origin means the graph passes through the origin?	No. xy = 1 is symmetric about the origin and misses it.	math precalculus graphs intercepts symmetry trap	Math::Graphs, intercepts and symmetry
Trap: -y = -x^3 + x is a different equation from y = x^3 - x, so no origin symmetry?	It is the same equation multiplied by -1, so it passes.	math precalculus graphs intercepts symmetry trap	Math::Graphs, intercepts and symmetry
Trap: can the graph of y = f(x) be symmetric about the x-axis?	Only y = 0. Otherwise one x would need two y's, y and -y.	math precalculus graphs intercepts symmetry trap	Math::Graphs, intercepts and symmetry
Trap: (0, 0) is either an x-intercept or a y-intercept, not both?	It is both: the origin is on both axes.	math precalculus graphs intercepts symmetry trap	Math::Graphs, intercepts and symmetry
Trap: only even powers of x, so y-axis symmetric. Always safe to skip the test?	No. It is a guess, and y = (x - 1)^2 fails: the powers are hidden until expanded.	math precalculus graphs intercepts symmetry trap	Math::Graphs, intercepts and symmetry
Slope of the line through (x1, y1) and (x2, y2)?	m = (y2 - y1)/(x2 - x1), rise over run, when x1 is not x2.	math precalculus lines slope slope	Math::Lines and slope
Why is the slope of a line the same for every pair of its points?	Each pair makes a right triangle with the same angles; similar triangles share the ratio rise/run.	math precalculus lines slope slope	Math::Lines and slope
Positive, negative, zero and undefined slope: what does each look like?	Rises left to right; falls; horizontal; vertical.	math precalculus lines slope slope	Math::Lines and slope
A slope of 3/4 means?	Up 3 for every 4 to the right.	math precalculus lines slope slope	Math::Lines and slope
Slope of a vertical line?	Undefined: the run is 0 and there is nothing to divide by.	math precalculus lines slope slope	Math::Lines and slope
Slope of a horizontal line?	0.	math precalculus lines slope slope	Math::Lines and slope
Point-slope form?	y - y1 = m(x - x1).	math precalculus lines slope forms	Math::Lines and slope
Slope-intercept form, and what b is?	y = mx + b. b is the y-intercept.	math precalculus lines slope forms	Math::Lines and slope
General form of a line?	Ax + By = C, with A and B not both 0.	math precalculus lines slope forms	Math::Lines and slope
Which form covers vertical lines?	Only the general form, as x = a. There is no y = mx + b with no slope.	math precalculus lines slope forms	Math::Lines and slope
Where does point-slope form come from?	The slope from (x1, y1) to any point (x, y) of the line is m: (y - y1)/(x - x1) = m.	math precalculus lines slope forms	Math::Lines and slope
Equation of a vertical line through (3, 7)?	x = 3.	math precalculus lines slope forms	Math::Lines and slope
Equation of a horizontal line through (3, 7)?	y = 7.	math precalculus lines slope forms	Math::Lines and slope
Parallel lines: slopes?	Equal slopes, different intercepts. Or both vertical.	math precalculus lines slope parallel-perpendicular	Math::Lines and slope
Perpendicular lines: slopes?	Product -1: each is the negative reciprocal of the other.	math precalculus lines slope parallel-perpendicular	Math::Lines and slope
Negative reciprocal of 2/3?	-3/2.	math precalculus lines slope parallel-perpendicular	Math::Lines and slope
Why does a perpendicular slope flip the fraction?	A quarter turn sends the step (a, b) to (-b, a), swapping run and rise.	math precalculus lines slope parallel-perpendicular	Math::Lines and slope
Slope of a line perpendicular to x = 5?	0: horizontal lines. The product rule does not apply, since x = 5 has no slope.	math precalculus lines slope parallel-perpendicular	Math::Lines and slope
Slope through (2, 3) and (-4, 5)?	-1/3.	math precalculus lines slope problems	Math::Lines and slope
Slope and y-intercept of 2x - 3y = 6?	Slope 2/3, y-intercept -2.	math precalculus lines slope problems	Math::Lines and slope
Line with slope 4 through (1, 2)?	y = 4x - 2.	math precalculus lines slope problems	Math::Lines and slope
Line through (1, 2) and (5, -3), general form?	5x + 4y = 13.	math precalculus lines slope problems	Math::Lines and slope
Line parallel to y = 2x + 1 through (3, 4)?	y = 2x - 2.	math precalculus lines slope problems	Math::Lines and slope
Line perpendicular to y = 2x + 1 through (3, 4)?	y = -x/2 + 11/2.	math precalculus lines slope problems	Math::Lines and slope
How can slopes show three points are on one line?	The slope from the first to the second equals the slope from the second to the third.	math precalculus lines slope problems	Math::Lines and slope
Trap: slope = (y2 - y1)/(x1 - x2)?	No. Both differences must start from the same point; mixing flips the sign.	math precalculus lines slope trap	Math::Lines and slope
Trap: slope = (x2 - x1)/(y2 - y1)?	No, that is run over rise. Slope is rise over run.	math precalculus lines slope trap	Math::Lines and slope
Trap: a vertical line has slope 0?	No, undefined. The horizontal line has slope 0.	math precalculus lines slope trap	Math::Lines and slope
Trap: y = 3x - 1 and 6x - 2y = 2 are parallel?	No, they are the same line: same slope and same intercept.	math precalculus lines slope trap	Math::Lines and slope
Trap: perpendicular to slope 2/3 is slope -2/3?	No. That is a mirror image. Perpendicular is -3/2: flip and change sign.	math precalculus lines slope trap	Math::Lines and slope
Trap: in y - 2 = 4(x + 1), the point is (1, 2)?	No. x + 1 = x - (-1), so the point is (-1, 2).	math precalculus lines slope trap	Math::Lines and slope
Trap: in 2x - 3y = 6, the slope is 2?	No. Solve for y first: y = (2/3)x - 2, slope 2/3.	math precalculus lines slope trap	Math::Lines and slope
What is a circle?	The set of all points at a fixed distance r, the radius, from a fixed point, the center.	math precalculus circles definition	Math::Circles
Standard form of a circle with center (h, k) and radius r?	(x - h)^2 + (y - k)^2 = r^2.	math precalculus circles standard-form	Math::Circles
Where does the standard form come from?	The distance formula from (x, y) to (h, k), set equal to r and squared.	math precalculus circles standard-form	Math::Circles
The unit circle's equation?	x^2 + y^2 = 1: center (0, 0), radius 1.	math precalculus circles standard-form	Math::Circles
Center and radius of (x - 3)^2 + (y + 1)^2 = 7?	Center (3, -1), radius the square root of 7.	math precalculus circles problems	Math::Circles
Center and radius of x^2 + (y - 2)^2 = 9?	Center (0, 2), radius 3.	math precalculus circles problems	Math::Circles
Standard form, center (-2, 3), radius 4?	(x + 2)^2 + (y - 3)^2 = 16.	math precalculus circles problems	Math::Circles
General form of a circle?	x^2 + y^2 + ax + by + c = 0, with x^2 and y^2 having the same coefficient.	math precalculus circles general-form	Math::Circles
How do you get center and radius from the general form?	Complete the square in x and in y.	math precalculus circles general-form	Math::Circles
How do you complete the square on x^2 + ax?	Add (a/2)^2: x^2 + ax + (a/2)^2 = (x + a/2)^2. Add the same to the other side.	math precalculus circles general-form	Math::Circles
x^2 + y^2 + 4x - 6y + 12 = 0: center and radius?	(x + 2)^2 + (y - 3)^2 = 1: center (-2, 3), radius 1.	math precalculus circles problems	Math::Circles
x^2 + y^2 - 4x + 6y - 3 = 0: center and radius?	(x - 2)^2 + (y + 3)^2 = 16: center (2, -3), radius 4.	math precalculus circles problems	Math::Circles
2x^2 + 2y^2 - 12x + 8y - 24 = 0: first step?	Divide by 2. Then center (3, -2), radius 5.	math precalculus circles problems	Math::Circles
After completing the square the right side is 0. What is the graph?	A single point, the center.	math precalculus circles general-form	Math::Circles
After completing the square the right side is negative. What is the graph?	Nothing: a sum of squares is never negative.	math precalculus circles general-form	Math::Circles
Is x^2 + 2y^2 = 4 a circle?	No: different coefficients on x^2 and y^2. It is an ellipse.	math precalculus circles general-form	Math::Circles
How do you tell if a point is inside a circle?	Its squared distance from the center is less than r^2.	math precalculus circles problems	Math::Circles
Circle with a diameter from (1, 4) to (-3, 2)?	Center the midpoint (-1, 3), r^2 = 5: (x + 1)^2 + (y - 3)^2 = 5.	math precalculus circles problems	Math::Circles
Circle with center (2, -3) through (5, 1)?	r^2 = 9 + 16 = 25: (x - 2)^2 + (y + 3)^2 = 25.	math precalculus circles problems	Math::Circles
How many whole-number points lie on x^2 + y^2 = 25?	12: the four on the axes and eight from 3-4-5.	math precalculus circles problems	Math::Circles
Why is (3/5, 4/5) on the unit circle?	9/25 + 16/25 = 1: the 3-4-5 triangle scaled by 1/5.	math precalculus circles problems	Math::Circles
How many times can a circle meet the x-axis?	Twice, once (touching), or never.	math precalculus circles problems	Math::Circles
Is a circle the graph of a function y = f(x)?	No. A vertical line through it meets it twice.	math precalculus circles definition	Math::Circles
Trap: (x + 2)^2 + (y - 3)^2 = 16 has center (2, -3)?	No, (-2, 3). x + 2 = x - (-2).	math precalculus circles trap	Math::Circles
Trap: (x - 1)^2 + (y - 1)^2 = 9 has radius 9?	No, 3. The right side is r^2.	math precalculus circles trap	Math::Circles
Trap: completing x^2 + 6x, add 6^2 = 36?	No. Add half of 6, squared: 9.	math precalculus circles trap	Math::Circles
Trap: complete the square on the left, leave the right side alone?	No. Whatever is added on the left must be added on the right.	math precalculus circles trap	Math::Circles
Trap: x^2 + y^2 + 4x - 6y + 14 = 0 is a small circle?	No. It becomes (x + 2)^2 + (y - 3)^2 = -1: no points at all.	math precalculus circles trap	Math::Circles
Trap: 3x^2 + 3y^2 - 6x = 0: complete the square on x^2 - 6x?	Divide by 3 first: x^2 + y^2 - 2x = 0, then (x - 1)^2 + y^2 = 1.	math precalculus circles trap	Math::Circles
Trap: to find r from the center and a point, take the square root and then square it again?	Unneeded. The equation wants r^2, which is the squared distance itself.	math precalculus circles trap	Math::Circles
What is a right triangle?	A triangle with one angle of 90 degrees, a square corner.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
Can a triangle have two right angles?	No. The three angles add up to 180 degrees, so two right angles would leave 0 degrees for the third.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
What is the hypotenuse?	The side opposite the right angle in a right triangle. It is always the longest side.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
What are the legs of a right triangle?	The two sides that form the right angle.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
What does the small square drawn in a corner of a triangle mean?	That the angle there is a right angle, 90 degrees.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
State the Pythagorean theorem.	In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c^2 = a^2 + b^2.	math geometry pythagorean-theorem theorem	Math::Pythagorean theorem
In c^2 = a^2 + b^2, which letter must be the hypotenuse?	c, the side opposite the right angle. a and b are the legs, in either order.	math geometry pythagorean-theorem theorem	Math::Pythagorean theorem
State the converse of the Pythagorean theorem.	If the square of one side of a triangle equals the sum of the squares of the other two, the triangle is a right triangle, and the right angle is opposite that side.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
What is the converse of a statement if A then B?	If B then A. It is a different claim and must be proved separately.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Give an example of a true statement whose converse is false.	If it is a dog, it has four legs. The converse, if it has four legs it is a dog, is false.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
What does the theorem let you do, and what does the converse let you do?	The theorem finds a missing side when you know the angle is right. The converse proves the angle is right when you know the three sides.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Testing three lengths for a right triangle: what is the first step?	Find the longest side. It is the only candidate for the hypotenuse.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Why check only the longest side as the hypotenuse?	A right angle would be opposite the hypotenuse, and the hypotenuse is always the longest side. With a leg in c's place, even a true right triangle fails.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Sides 6, 4, 3: right triangle?	No. The longest is 6, and 36 is not 16 + 9 = 25.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Sides 5, 12, 13: right triangle?	Yes. 13^2 = 169 = 25 + 144. The hypotenuse is 13.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Sides 7, 24, 25: right triangle?	Yes. 25^2 = 625 = 49 + 576. The hypotenuse is 25.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Sides 4, 5, 6: right triangle?	No. 36 is less than 16 + 25 = 41, so the largest angle is acute.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Sides 2, 2, 3: right triangle?	No. 9 is more than 4 + 4 = 8, so the largest angle is obtuse.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Legs 4 and 3: hypotenuse?	5, since 16 + 9 = 25.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Legs 5 and 12: hypotenuse?	13, since 25 + 144 = 169.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Legs 14 and 48: hypotenuse?	50, since 196 + 2304 = 2500. It is 7-24-25 doubled.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Hypotenuse 13 and one leg 5: the other leg?	12, since 169 - 25 = 144. Subtract from the hypotenuse's square.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Legs 1 and 1: hypotenuse?	The square root of 2, about 1.4142. Exact only as a square root.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
If c^2 is less than a^2 + b^2, with c the longest side, what is the angle opposite c?	Acute, less than 90 degrees.	math geometry pythagorean-theorem acute-obtuse	Math::Pythagorean theorem
If c^2 is greater than a^2 + b^2, with c the longest side, what is the angle opposite c?	Obtuse, more than 90 degrees.	math geometry pythagorean-theorem acute-obtuse	Math::Pythagorean theorem
What is the triangle inequality?	Each side of a triangle is shorter than the other two added together. If not, the three lengths make no triangle.	math geometry pythagorean-theorem acute-obtuse	Math::Pythagorean theorem
Sides 3, 4, 7: what kind of triangle?	None. 3 + 4 = 7, so the three lengths lie flat along a line.	math geometry pythagorean-theorem acute-obtuse	Math::Pythagorean theorem
Which general law has the Pythagorean theorem as its 90 degree case?	The law of cosines, c^2 = a^2 + b^2 - 2ab cos C. At C = 90 degrees cos C = 0.	math geometry pythagorean-theorem acute-obtuse	Math::Pythagorean theorem
Rearrangement proof: what does a square of side a + b contain?	Four copies of the right triangle, area ab/2 each, and a tilted square of side c in the middle.	math geometry pythagorean-theorem proof	Math::Pythagorean theorem
Finish the rearrangement proof from (a + b)^2 = 4(ab/2) + c^2.	Expand: a^2 + 2ab + b^2 = 2ab + c^2. Subtract 2ab: a^2 + b^2 = c^2.	math geometry pythagorean-theorem proof	Math::Pythagorean theorem
What is a Pythagorean triple?	Three whole numbers a, b, c with a^2 + b^2 = c^2, such as 3, 4, 5.	math geometry pythagorean-theorem triples	Math::Pythagorean theorem
Name five Pythagorean triples worth knowing by sight.	3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29.	math geometry pythagorean-theorem triples	Math::Pythagorean theorem
If 3, 4, 5 is a triple, why is 6, 8, 10?	Multiply every side by k and every square by k^2, so the equation still holds.	math geometry pythagorean-theorem triples	Math::Pythagorean theorem
From c^2 = 25, why is c = 5 and not -5?	c is a length, and lengths are not negative. The square root sign means the non-negative root.	math geometry pythagorean-theorem theorem	Math::Pythagorean theorem
Burj Khalifa example: which angle is the right angle?	Where the line of sight touches the Earth: a tangent line meets the radius at 90 degrees.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Burj Khalifa example: why is (R + h)^2 - R^2 risky on a calculator?	It subtracts two nearly equal numbers, about 15.7 million each, which cancels their leading digits. Use 2Rh + h^2 instead.	math geometry pythagorean-theorem precision	Math::Pythagorean theorem
How far can you see from 1483 feet up, with Earth's radius 3960 miles?	About 47 miles: d is the square root of 2Rh + h^2, about 2224.58 square miles.	math geometry pythagorean-theorem problems	Math::Pythagorean theorem
Trap: legs 3 and 4, so c = 3 + 4 = 7?	No. Squares add, not sides: c^2 = 9 + 16 = 25, c = 5. 3 + 4 = 7 would be a flat line.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: hypotenuse 10, leg 6, so the other leg is the square root of 100 + 36?	No. Subtract when c is known: 100 - 36 = 64, leg 8. Adding would make a leg longer than the hypotenuse.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: sides 8, 15, 17 listed as 17, 8, 15. Test 15^2 against 17^2 + 8^2?	No. Sort first: 17 is longest. 289 = 64 + 225, so it is right, with hypotenuse 17.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: c^2 = a^2 + b^2 holds for every triangle if you pick the right labels?	No. It holds only for right triangles. For 4, 5, 6 no labelling works.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: is the square root of (a^2 + b^2) equal to a + b?	No. The square root of 9 + 16 is 5, not 7. A square root does not split over a sum.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: a triangle has sides 1, 1 and the square root of 2. Which angle is right?	The one opposite the square root of 2, the longest side: 1 + 1 = 2.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: every triangle with c^2 greater than a^2 + b^2 is obtuse. True for any labels?	Only when c is the longest side. With a short side as c, the inequality says nothing about the largest angle.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: the theorem proves a triangle with sides 5, 12, 13 is right. Theorem or converse?	The converse. The theorem assumes the right angle; the converse concludes it from the sides.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
What does perpendicular mean?	Two lines meeting at a right angle; all four angles where they cross are 90 degrees. Written with the symbol that looks like an upside-down T.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
A right angle in degrees, radians and gradians?	90 degrees, pi/2 radians, 100 gradians: a quarter turn in each.	math geometry pythagorean-theorem vocabulary	Math::Pythagorean theorem
The two acute angles of a right triangle add up to what?	90 degrees: they are complementary. If one is theta, the other is 90 - theta.	math geometry pythagorean-theorem theorem	Math::Pythagorean theorem
Why is the hypotenuse the longest side?	The largest angle of a triangle is opposite the longest side, and the right angle is the largest.	math geometry pythagorean-theorem theorem	Math::Pythagorean theorem
Why is the hypotenuse shorter than the sum of the legs?	(a + b)^2 = a^2 + b^2 + 2ab, which is more than c^2 by 2ab.	math geometry pythagorean-theorem theorem	Math::Pythagorean theorem
Side ratios of a 45-45-90 triangle, and where they come from?	1 : 1 : square root of 2. Half a square cut along its diagonal: c^2 = 1 + 1.	math geometry pythagorean-theorem special-triangles	Math::Pythagorean theorem
Side ratios of a 30-60-90 triangle, and where they come from?	1 : square root of 3 : 2. Half an equilateral triangle of side 2: long leg^2 = 4 - 1 = 3.	math geometry pythagorean-theorem special-triangles	Math::Pythagorean theorem
In a 30-60-90 triangle, which side is half the hypotenuse?	The short leg, opposite the 30 degree angle.	math geometry pythagorean-theorem special-triangles	Math::Pythagorean theorem
45-45-90 with legs 7: hypotenuse?	7 times the square root of 2, about 9.90.	math geometry pythagorean-theorem special-triangles	Math::Pythagorean theorem
30-60-90 with hypotenuse 10: legs?	5 and 5 times the square root of 3, about 8.66.	math geometry pythagorean-theorem special-triangles	Math::Pythagorean theorem
Space diagonal of a box p by q by r?	The square root of p^2 + q^2 + r^2: Pythagoras on a face, then again with the third edge.	math geometry pythagorean-theorem three-d	Math::Pythagorean theorem
Space diagonal of a 2 by 3 by 6 box?	7, since 4 + 9 + 36 = 49.	math geometry pythagorean-theorem three-d	Math::Pythagorean theorem
Diagonal of a unit cube?	The square root of 3.	math geometry pythagorean-theorem three-d	Math::Pythagorean theorem
State the law of cosines.	c^2 = a^2 + b^2 - 2ab cos C, for any triangle, with C the angle opposite c.	math geometry pythagorean-theorem law-of-cosines	Math::Pythagorean theorem
How does the law of cosines reduce to Pythagoras?	At C = 90 degrees cos C = 0, so the last term vanishes.	math geometry pythagorean-theorem law-of-cosines	Math::Pythagorean theorem
Trap: Pythagoras is a consequence of the law of cosines?	No, the other way round. The law of cosines is usually proved with Pythagoras; Pythagoras is the special case, not the consequence.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: in a 30-60-90 triangle, the long leg is half the hypotenuse?	No. The short leg, opposite 30 degrees, is half. The long leg is the square root of 3 times the short leg.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: the hypotenuse is the denominator of every basic trig ratio?	No. Tangent is opposite over adjacent. Sine and cosine divide by the hypotenuse.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Why is the converse of the Pythagorean theorem true?	Build a right triangle with legs a and b. Its hypotenuse is the square root of a^2 + b^2 = c, so both triangles have sides a, b, c, and SSS makes them congruent: the right angle carries over.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Which congruence rule proves the converse of the Pythagorean theorem?	Side-Side-Side.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Trap: the right angle is opposite whichever side is alone on one side of the equation?	Only if you wrote it as a^2 + b^2 = c^2. In c^2 - b^2 = a^2 a leg is alone. Safe rule: the right angle is opposite the longest side.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
In a^2 + b^2 = c^2, is c a name or a job?	A job: the side opposite the right angle. The equation says (leg)^2 + (other leg)^2 = (hypotenuse)^2.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
In the Pythagorean equation, which side stands alone and which stand together?	The longest side, squared, stands alone. The other two, squared and added, stand together on the other side.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Does it matter whether c^2 is on the left or the right of the equals sign?	No. a^2 + b^2 = c^2 and c^2 = a^2 + b^2 are the same equation read both ways. What matters is that c^2 is alone.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
The three steps of the Pythagorean test?	1. Find the longest side, c. 2. Put c^2 alone on one side, the other two squares added on the other. 3. Compare.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
Why must the side standing alone be the longest?	Two squares added are bigger than either one alone, so the lone square must be the biggest.	math geometry pythagorean-theorem converse	Math::Pythagorean theorem
3-4-5 with 4 standing alone: 3^2 + 5^2 against 4^2?	34 against 16: not equal. Same right triangle, wrong setup: it claims leg^2 + hypotenuse^2 = other leg^2.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
3-4-5 with 3 standing alone: 4^2 + 5^2 against 3^2?	41 against 9: not equal. A short side alone always leaves the lone square too small.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: 3^2 + 5^2 = 34 is not 4^2 = 16, so the 3-4-5 triangle is not right?	No. The triangle is right; the setup is wrong. The longest side, 5, must stand alone: 9 + 16 = 25.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Trap: c^2 = a^2 + b^2 is wrong because c must be on the right?	No. An equation reads both ways. Only alone and together matter, not left and right.	math geometry pythagorean-theorem trap	Math::Pythagorean theorem
Rectangle, length l and width w: perimeter and area?	Perimeter 2l + 2w. Area lw.	math geometry area-volume formula	Math::Area and volume formulas
Triangle, base b and altitude h: area?	One half b h.	math geometry area-volume formula	Math::Area and volume formulas
Circle of radius r: circumference and area?	Circumference 2 pi r, which is pi d. Area pi r^2.	math geometry area-volume formula	Math::Area and volume formulas
Closed box l by w by h: volume and surface area?	Volume lwh. Surface 2lh + 2wh + 2lw, three pairs of matching rectangles.	math geometry area-volume formula	Math::Area and volume formulas
Sphere of radius r: volume and surface area?	Volume (4/3) pi r^3. Surface 4 pi r^2.	math geometry area-volume formula	Math::Area and volume formulas
Closed right circular cylinder, radius r and height h: volume and surface area?	Volume pi r^2 h. Surface 2 pi r^2 + 2 pi r h: two circles and the side.	math geometry area-volume formula	Math::Area and volume formulas
What is the altitude of a triangle?	The height measured at a right angle to the base, from the base to the opposite corner. Not a side, unless the triangle is right.	math geometry area-volume formula	Math::Area and volume formulas
Why is the area of a triangle half of bh?	A triangle is half of a rectangle with the same base and height.	math geometry area-volume formula	Math::Area and volume formulas
Unrolled, the side of a cylinder is what shape, and how big?	A rectangle 2 pi r wide and h tall, so its area is 2 pi r h.	math geometry area-volume formula	Math::Area and volume formulas
How do you tell a length, an area and a volume formula apart without memorising?	Count the lengths multiplied in each term: one is a length, two an area, three a volume. Constants do not count.	math geometry area-volume dimension	Math::Area and volume formulas
Double every length of a shape. What happens to perimeter, area, volume?	Perimeter times 2, area times 4, volume times 8.	math geometry area-volume dimension	Math::Area and volume formulas
Scale every length by k. What happens to an area and to a volume?	Area times k^2, volume times k^3.	math geometry area-volume dimension	Math::Area and volume formulas
Every side of a box is tripled. Volume?	27 times as large, 3^3.	math geometry area-volume dimension	Math::Area and volume formulas
A formula for a volume grows by 4 when lengths double. What is wrong?	It is an area in disguise: a volume must grow by 8.	math geometry area-volume dimension	Math::Area and volume formulas
Why do the constants 2, one half, pi and 4/3 not count in the dimension test?	They are pure numbers without units, so they cannot change feet into square feet.	math geometry area-volume dimension	Math::Area and volume formulas
Which fact did Archimedes prove about the sphere's surface?	It equals the side of the cylinder that just fits around it: 2 pi r times 2r = 4 pi r^2.	math geometry area-volume formula	Math::Area and volume formulas
How is the surface area of a sphere related to its volume?	It is what the volume gains per unit of radius: a skin of thickness t has volume about 4 pi r^2 t.	math geometry area-volume dimension	Math::Area and volume formulas
Circle of radius 5 m: area and circumference?	25 pi, about 78.54 square meters. 10 pi, about 31.42 meters.	math geometry area-volume problems	Math::Area and volume formulas
Sphere of radius 3 ft: volume and surface area?	36 pi cubic feet and 36 pi square feet: the same number in different units.	math geometry area-volume problems	Math::Area and volume formulas
Cylinder r 9, h 8 against r 8, h 9: which holds more?	r 9, h 8: 648 pi against 576 pi. The radius is squared, the height is not.	math geometry area-volume problems	Math::Area and volume formulas
Square of side 2 inside a circle, corners on the circle: area of the circle?	2 pi. The diagonal is the square root of 8, so r^2 = 8/4 = 2.	math geometry area-volume problems	Math::Area and volume formulas
Circle inside a square of side 2, touching all sides: area outside the circle?	4 - pi, about 0.86.	math geometry area-volume problems	Math::Area and volume formulas
Square of side 6 with a border 2 wide all round: area of the border?	64. The outer side is 6 + 2 + 2 = 10, and 100 - 36 = 64.	math geometry area-volume problems	Math::Area and volume formulas
Wheel of diameter 16 in, 4 turns: distance?	64 pi inches, about 201 inches. One turn is pi d.	math geometry area-volume problems	Math::Area and volume formulas
Disk of diameter 4 ft rolled 20 ft: how many turns?	20 / (4 pi) = 5/pi, about 1.59.	math geometry area-volume problems	Math::Area and volume formulas
Norman window: rectangle 4 wide and 6 tall, semicircle on top. Area?	24 + 2 pi, about 30.28 square feet. The semicircle has radius 2, area half of 4 pi.	math geometry area-volume problems	Math::Area and volume formulas
Ornament: triangle base 4, height 6, semicircle of diameter 4 on top. Area?	12 + 2 pi, about 18.28 square centimeters.	math geometry area-volume problems	Math::Area and volume formulas
Trap: the volume of a sphere is 4 pi r^3?	No. (4/3) pi r^3. 4 pi r^3 is bigger than the 8r^3 cube around the sphere.	math geometry area-volume trap	Math::Area and volume formulas
Trap: the surface area of a sphere is (4/3) pi r^2?	No. 4 pi r^2. The dimension test cannot catch this one: only the constant is wrong.	math geometry area-volume trap	Math::Area and volume formulas
Trap: a wheel of diameter d rolls 2 pi d in one turn?	No, pi d. The 2 in 2 pi r is there because r is half of d.	math geometry area-volume trap	Math::Area and volume formulas
Trap: triangle with base 10 and two slanted sides 13. Area one half 10 times 13?	No. The altitude is 12, from 13^2 - 5^2 = 144, so the area is 60.	math geometry area-volume trap	Math::Area and volume formulas
Trap: a 16 inch pizza for twice the price of a 10 inch one is a bad deal?	No. Area grows with the square: 2.56 times the pizza for 2 times the price.	math geometry area-volume trap	Math::Area and volume formulas
Trap: a circle of radius 2 has area equal to its circumference, both 4 pi. Are they equal?	Only as numbers. One is square units, the other units; change the unit and they differ.	math geometry area-volume trap	Math::Area and volume formulas
Trap: a closed can and an open can of the same size have the same surface area?	No. The open can is missing one lid, pi r^2.	math geometry area-volume trap	Math::Area and volume formulas
Trap: border 2 wide around a square of side 6 makes an outer square of side 8?	No, 10. The border is on both sides.	math geometry area-volume trap	Math::Area and volume formulas
What does it mean for two triangles to be congruent?	Each pair of corresponding angles is equal and each pair of corresponding sides is equal: the same triangle in two places.	math geometry triangles definition	Math::Congruent and similar triangles
What does it mean for two triangles to be similar?	Corresponding angles are equal and corresponding sides are proportional: the same shape, possibly a different size.	math geometry triangles definition	Math::Congruent and similar triangles
What is the scale factor of two similar triangles?	The one ratio every pair of corresponding sides shares, k = d/a = e/b = f/c.	math geometry triangles definition	Math::Congruent and similar triangles
Congruent triangles are similar with what scale factor?	k = 1.	math geometry triangles definition	Math::Congruent and similar triangles
How do you tell which sides correspond?	By the angles they sit between, or the angle they are opposite, never by where they are drawn.	math geometry triangles definition	Math::Congruent and similar triangles
Symbols for congruent and for similar?	Congruent: the equals sign with a tilde above it. Similar: a tilde.	math geometry triangles definition	Math::Congruent and similar triangles
State the SSS case for congruence.	If all three pairs of corresponding sides are equal, the triangles are congruent.	math geometry triangles congruence	Math::Congruent and similar triangles
State the SAS case for congruence.	Two pairs of sides equal and the angles between them equal: congruent.	math geometry triangles congruence	Math::Congruent and similar triangles
State the ASA case for congruence.	Two pairs of angles equal and the sides between them equal: congruent.	math geometry triangles congruence	Math::Congruent and similar triangles
Why do SSS, SAS and ASA prove congruence?	Each set of three data allows only one triangle to be built, so two triangles with the same data are copies of it.	math geometry triangles congruence	Math::Congruent and similar triangles
Which of ASA, SAS, AAA, SSS is not a congruence case?	AAA. Equal angles fix the shape only, not the size.	math geometry triangles congruence	Math::Congruent and similar triangles
Why is SSA, two sides and an angle not between them, not a congruence case?	The second side can swing and meet the base in two places, so two different triangles can fit the same data.	math geometry triangles congruence	Math::Congruent and similar triangles
When does SSA work after all?	When the angle is 90 degrees: hypotenuse and leg fix a right triangle, by Pythagoras.	math geometry triangles congruence	Math::Congruent and similar triangles
State the AA case for similarity.	If two pairs of corresponding angles are equal, the triangles are similar.	math geometry triangles similarity	Math::Congruent and similar triangles
Why is AA enough, rather than AAA?	The angles add to 180 degrees, so the third is forced.	math geometry triangles similarity	Math::Congruent and similar triangles
State the SSS case for similarity.	If all three pairs of sides are in the same ratio, the triangles are similar.	math geometry triangles similarity	Math::Congruent and similar triangles
State the SAS case for similarity.	Two pairs of sides in the same ratio, and the angles between them equal: similar.	math geometry triangles similarity	Math::Congruent and similar triangles
Similar triangles, scale factor k: ratio of perimeters and of areas?	Perimeters k, areas k^2.	math geometry triangles similarity	Math::Congruent and similar triangles
Once two triangles are known similar, how do you find the angles and a missing side?	Copy the angles across. For the side, one proportion with the scale factor.	math geometry triangles similarity	Math::Congruent and similar triangles
Why is the slope of a line the same whichever two points you use?	Any two points make a right triangle with the same angles, so the triangles are similar and rise over run is one ratio.	math geometry triangles similarity	Math::Congruent and similar triangles
Example 5: sides 3 and 6 in one triangle; the side matching 3 is 5 in the other. Find x matching 6.	k = 5/3, x = 6 times 5/3 = 10.	math geometry triangles problems	Math::Congruent and similar triangles
Right triangle 30-60-90 with short leg 2, hypotenuse 4. Similar one with hypotenuse 8: short leg?	4. The scale factor is 2.	math geometry triangles problems	Math::Congruent and similar triangles
Sides 20 and 45 in one triangle, and 30 matching the 20 in a similar one. Side matching 45?	67.5. k = 3/2.	math geometry triangles problems	Math::Congruent and similar triangles
Sides 3, 2 with 120 degrees between, and 4, 3 with 120 degrees between. Similar?	No. 4/3 is not 3/2, so SAS similarity fails.	math geometry triangles problems	Math::Congruent and similar triangles
Two triangles with angles 25 and 100 degrees each. Similar?	Yes, by AA.	math geometry triangles problems	Math::Congruent and similar triangles
A 6 ft person casts a 4 ft shadow; a tree casts 30 ft. Height of the tree?	45 ft. Similar triangles by AA, k = 7.5.	math geometry triangles problems	Math::Congruent and similar triangles
Trap: two triangles with all three angles equal are congruent?	No, only similar. A triangle and its double have the same angles.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: all right triangles are similar?	No. One right angle is one angle; similarity needs two. 3-4-5 is not similar to 1-1-root 2.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: all equilateral triangles are similar?	Yes: every angle is 60 degrees, so AA holds. Not every trap card has the answer no.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: the scale factor is 3, so the area is 3 times as big?	No, 9 times: area scales by k^2.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: match the top side of one figure with the top side of the other?	No. Match by angles; the book draws similar triangles in different orientations on purpose.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: two sides equal and one angle equal means congruent?	Only if the angle is between the two sides (SAS). Otherwise it is SSA and two triangles may fit.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: two angles and a side equal means congruent, whichever side?	Yes, as long as it is the corresponding side: the third angle is forced, so it becomes ASA.	math geometry triangles trap	Math::Congruent and similar triangles
Trap: two triangles have equal areas, so they are congruent?	No. Base 4 height 3 and base 6 height 2 both have area 6 and differ in shape.	math geometry triangles trap	Math::Congruent and similar triangles