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:
$\angle DBC$ is a right angle

Answer 1

Given : Right triangle ABC, right angle is at C, M is mid- point of hypotenuse AB and $DM=CM$

To show: $\angle DBC$ is a right angle

Proof :

in triangle $DBM$ and triangle $AMC$,

$DM=MC$ ( Given )

$BM=MA$ ( Given )

$\angle DMB=\angle AMC$ (Vertically opposite angle)

Therefore, by SAS criterion of congruency, triangle $DMB$ is congruent to triangle $AMC$ So,

$DB=AC$ (By CPCT)

$\angle$BDM= $\angle CAM$ (by CPCT)

Now, in triangle $DBC$ and triangle $ABC$,

$DB=AC$ ( proved above)

$BC=BC$ ( common)

$DM=MA,MC=MB$ ( proved above)

so $DM+MC=MA+MB$

$DC=AB$

Therefore, by SSS criterion of congruency, triangle $DBC$ is congruent to triangle $ACB$.

Therefore,

$\angle DBC=\angle ACB={90}^{\circ }$ (By CPCT)