Question

right triangle abc is isosceles and point m is the midpoint of the hypotenuse.
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.

Explanation:

Step1: Identify triangle properties

$\triangle ABC$ is isosceles right triangle, so $AB=BC$, $\angle ABC=90^\circ$, hypotenuse $AC$. $M$ is midpoint of $AC$.

Step2: Apply right triangle theorem

In a right triangle, the midpoint of the hypotenuse is equidistant from all 3 vertices: $AM=BM=MC$. Since $\triangle ABC$ is isosceles, $AM=\frac{1}{2}AC$, and $AB=BC=\frac{AC}{\sqrt{2}}$.

Step3: Analyze $\triangle AMB$

$AM=BM$, so $\triangle AMB$ is isosceles. $\angle BAM=45^\circ$, $\angle ABM=45^\circ$, so $\angle AMB=90^\circ$. Thus it is an isosceles right triangle.

Step4: Eliminate other options

• $\triangle AMB$ is smaller than $\triangle ABC$, so not congruent.
• It has a right angle, not obtuse.
• It has two equal sides, not scalene.

Answer:

It is an isosceles right triangle.