Asked by martin
Right triangle ABC is isosceles and point M is the midpoint of the hypotenuse.
Right triangle A B C is shown. Point M is the midpoint of the hypotenuse. The lengths of sides A B and B C are congruent. The lengths of line segments A M and M C are congruent.
What is true about triangle AMB?
It is congruent to triangle ABC.
It is an obtuse triangle.
It is an isosceles right triangle.
It is a scalene triangle.
All Answers 1
Answered by
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It is an isosceles right triangle.
Reason: Let B be the right angle so AC is the hypotenuse and M its midpoint. In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices, so AM = BM. Also, with A = (1,0), B = (0,0), C = (0,1) (an isosceles right triangle), M = (1/2,1/2) and the vectors MA = (1/2,-1/2) and MB = (-1/2,-1/2) have dot product 0, so ∠AMB = 90°. Thus triangle AMB is both isosceles (AM = BM) and right.
Reason: Let B be the right angle so AC is the hypotenuse and M its midpoint. In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices, so AM = BM. Also, with A = (1,0), B = (0,0), C = (0,1) (an isosceles right triangle), M = (1/2,1/2) and the vectors MA = (1/2,-1/2) and MB = (-1/2,-1/2) have dot product 0, so ∠AMB = 90°. Thus triangle AMB is both isosceles (AM = BM) and right.
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