12 multiple-choice questions, progressively harder.
M is the midpoint of AB at (1,−2)(1,-2)(1,−2). Endpoint A is (s,1)(s,1)(s,1) and AB is 2342\sqrt{34}234 units. Which list gives all possible values of sss?
Solution
Correct answer: D
The distance from A to M is half of AB, so its square is 34.
The horizontal change is −5-5−5 or 5, giving s=−4s=-4s=−4 or s=6s=6s=6.
M is the midpoint of AB at (2,1)(2,1)(2,1), and N is the midpoint of AC at (−1,3)(-1,3)(−1,3). If A=(4,−2)A=(4,-2)A=(4,−2), what is BC?
Correct answer: B
Recover both endpoints from their midpoint data.
For AAA at (−2,−1)(-2,-1)(−2,−1) and BBB at (4,3)(4,3)(4,3), point PPP at (−1,4)(-1,4)(−1,4) is equally far from A and B. Is P the midpoint of AB?
Correct answer: C
The squared distances from P to the endpoints are both 26, but midpoint requires the point to be on AB too.
P differs from this point, so it is not the midpoint.
Triangle ABC has vertices (0,0)(0,0)(0,0), (4,1)(4,1)(4,1), and (1,5)(1,5)(1,5) in that order. Triangle DEF has vertices (7,−1)(7,-1)(7,−1), (6,3)(6,3)(6,3), and (2,0)(2,0)(2,0) in that order. Which conclusion is correct?
Correct answer: A
Compute the three corresponding side squares.
All three named pairs match. SSS proves ABC congruent to DEF. Swapping E and F would mismatch the unequal sides from A and D.
A has coordinates (−3,2)(-3,2)(−3,2). Point B has y-coordinate 5, and AB is 58\sqrt{58}58 units. Which point can be the midpoint of AB?
The vertical gap is 3.
Thus B has x-coordinate 4 or −10-10−10. Their midpoint x-coordinates are 1/21/21/2 or −13/2-13/2−13/2, and both midpoint heights are 7/27/27/2. Only the stated pair is offered.
A segment has midpoint (3,−1)(3,-1)(3,−1) and length 8 units. Both endpoints have y-coordinate −1-1−1. Which unordered pair of endpoints fits?
The common height makes the segment horizontal. Each endpoint is 4 units from the midpoint.
Both keep height −1-1−1.
Endpoints A and B are distinct and have midpoint M. If AM2=18AM^2=18AM2=18, what is AB2AB^2AB2?
The whole length is twice the half-length. Squaring multiplies the square by 4.
A is at (2,−4)(2,-4)(2,−4) and B is at (t,1)(t,1)(t,1). Which prescribed length makes the endpoint conditions impossible?
The fixed vertical gap already has square 25.
A length with square 24 is impossible. Square 25 gives one position; either larger offered square gives two.
The midpoint of AB is (1,2)(1,2)(1,2). Endpoint A is (a,−1)(a,-1)(a,−1), and B has x-coordinate −4-4−4. What is AB?
The midpoint conditions determine both unknown coordinates.
The endpoint gaps are −10-10−10 and 6.
Triangle ABC has squared side lengths 13 for AB, 50 for AC, and 41 for BC. A congruent triangle has squared side lengths 41 for DE, 13 for DF, and 50 for EF. Which vertex corresponds to B?
Vertex B meets the sides whose squares are 13 and 41. In DEF, only D meets the sides with those squares.
The vertex correspondence is A to F, B to D, C to E. All three squared lengths differ, so the match is unique.
A segment AB has midpoint (0,1)(0,1)(0,1). A lies on the x-axis and has positive x-coordinate, while AB has length 2172\sqrt{17}217. Which endpoint is B?
Write A as (a,0)(a,0)(a,0). The midpoint forces B to be (−a,2)(-a,2)(−a,2).
The positive restriction gives a=4a=4a=4, so B is (−4,2)(-4,2)(−4,2).
M is the midpoint of AB at (0,2)(0,2)(0,2). A lies on the x-axis and AB is 4 units long. How many ordered endpoint pairs (A,B)(A,B)(A,B) satisfy all conditions?
If A is (a,0)(a,0)(a,0), the midpoint forces B to be (−a,4)(-a,4)(−a,4).
Only a=0a=0a=0 works. Thus A is (0,0)(0,0)(0,0) and B is (0,4)(0,4)(0,4), giving one ordered pair.
Reset this practice set?
This clears every answer you have given and starts the set again from question 1.