12 multiple-choice questions, progressively harder.
For AAA at (−6,4)(-6,4)(−6,4) and BBB at (2,−1)(2,-1)(2,−1), what is the horizontal leg length in a coordinate right triangle?
Solution
Correct answer: B
The horizontal leg uses only the x-coordinate gap.
The y-coordinate gap is a separate leg.
The endpoints of a segment are (−1,−5)(-1,-5)(−1,−5) and (4,−3)(4,-3)(4,−3). What is its exact length?
Correct answer: D
The gaps are 5 horizontally and 2 vertically.
Which coordinate is shared by every point on the segment from (−4,−6)(-4,-6)(−4,−6) to (−4,2)(-4,2)(−4,2)?
Correct answer: A
This segment is vertical because both endpoints have x-coordinate −4-4−4.
The y-coordinate varies along the segment.
What is the midpoint of the segment from (−7,2)(-7,2)(−7,2) to (3,6)(3,6)(3,6)?
Correct answer: C
Average the matching coordinates.
A segment has endpoints (6,−7)(6,-7)(6,−7) and (6,−2)(6,-2)(6,−2). What is its length?
The x-coordinate does not change, so use the size of the vertical gap.
Which construction locates the midpoint of a slanted segment ABABAB?
A dilation centered at A keeps its image on ray AB and halves the distance from A.
The remaining segment MB is also half of AB, so the image is the midpoint.
Point MMM at (0,−2)(0,-2)(0,−2) is the midpoint of ABABAB. Endpoint AAA is at (−5,1)(-5,1)(−5,1). Find BBB.
Double each midpoint coordinate, then subtract the known endpoint coordinate.
The endpoints are (−3,−4)(-3,-4)(−3,−4) and (2,1)(2,1)(2,1). Which expression gives their straight-line distance?
The horizontal and vertical gaps are each 5. They are perpendicular legs.
Take the nonnegative square root to obtain the distance.
The endpoints of ABABAB are (2,5)(2,5)(2,5) and (8,5)(8,5)(8,5). Where is its midpoint?
The segment stays at height 5. Average its horizontal coordinates.
The midpoint is (5,5)(5,5)(5,5).
Why does swapping the endpoints leave the distance formula unchanged?
Reversing one difference changes its sign.
The same holds vertically, so the squared distance and its nonnegative root stay the same.
What is the midpoint of the segment joining (−4,−7)(-4,-7)(−4,−7) and (1,−2)(1,-2)(1,−2)?
The coordinate sums are −3-3−3 and −9-9−9. Divide each by 2.
A midpoint need not have integer coordinates.
Two nondegenerate triangles have three corresponding squared side lengths 131313, 343434, and 454545. What follows?
The three exact squared lengths match in the stated correspondence. Taking nonnegative roots supplies each equal side pair.
Applying the same step to the other two pairs gives SSS. No right angle follows from these data.
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