Question

In right triangle $ABC$, right angle is 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:
$△AMC\cong △BMD$

Answer 1

In right angled $△ABC$,

$\angle C={90}^{\circ }$,

$M$ is the mid-point of $AB$ i.e, $AM=MB$ & $DM=CM$

In $△AMC$ and $△BMD$,

$AM=BM$ ------ (M is the mid-point)

$\angle CMA=\angle DMB$ ------ (Vertically opposite angles)

$CM=DM$ ------ (Given)

$\therefore △AMC\cong △BMD$, by SAS Postulate