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#deck:Math::Midpoint formula
#tags:math precalculus midpoint-formula
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State the midpoint formula.	M = ((x1 + x2)/2, (y1 + y2)/2): the average of the x's and the average of the y's.	formula
The midpoint formula in three words?	Sum over two, for each coordinate.	formula
Each coordinate of the midpoint is what kind of average?	The arithmetic mean of the two coordinates.	formula
On a number line, the point halfway between a and b?	(a + b)/2.	formula
Halfway along a segment means which two things?	Equally far from both ends, and on the segment.	why
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.	why
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.	why
How far is the midpoint from each end?	Half the length of the segment.	why
Does the midpoint depend on which point is called P1?	No: x1 + x2 = x2 + x1.	why
The point a fraction t of the way from P1 to P2?	P1 + t(P2 - P1). t = 1/2 is the 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.	why
Given one end P1 and the midpoint M, the other end?	P2 = 2M - P1, coordinate by coordinate.	formula
Midpoint of (-5, 5) and (3, 1)?	(-1, 3).	problems
Midpoint of (1.5, -2) and (-3.5, 4)?	(-1, 1).	problems
One end (1, 2), midpoint (4, -1): other end?	(7, -4).	problems
One end (0, 0), midpoint (3, 5): other end?	(6, 10).	problems
The center of a circle from the two ends of a diameter?	The midpoint of the diameter.	problems
How can midpoints test for a parallelogram?	Its diagonals bisect each other: they have the same midpoint.	problems
The average of three points, and its name for a triangle?	The centroid, ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).	problems
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).	trap
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).	trap
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.	trap
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.	trap
Trap: which formulas use sums and which use differences?	Midpoint: sums. Distance and slope: differences.	trap