Question

In the right triangle $ABC$, right angled at $C$, $M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM=CM$. Point $D$ is joined to point $B$. Show that $△AMC$ is congruent to $△BMD$.

Answer 1

Step $1:$ Describe the figure.

The figure describing the given data as follows:

Step $2:$ Prove that $△AMC$ is Congruence to $△BMD$.

Since, $M$ is the mid-point of hypotenuse $AB$.

Therefore, $AM=BM$

Also, it is given that $DM=CM$.

$\angle AMC=\angle BMD$ ($\because$Vertically opposite angles)

By Side-angle-Side property $\left(SAS\right)$, $△AMC$ is Congruent to $△BMD$.

Hence proved, $△AMC$ is congruent to $△BMD$.