Question

State whether it is true/false:

In the given figure, $AB=AC$, then $AP=AQ$.

• A. True
• B. False

Answer 1

The correct option is A True

In triangle $ABC$, since $AB=AC$
thus, $\angle ABC=\angle ACB$ ....$\left(I\right)$ (angles opposite to equal sides are equal).

Now, in triangles $DPB$ and $DQC$,
$\angle P=\angle Q={90}^o$ (Given)
$\angle ABC=\angle ACB$ (From $\left(I\right)$)
$BD=DC$ (Given)
Thus triangles, $DPB\cong DQC$ (By $ASA$ Congruency criterion)
Hence, $DP=DQ$ (By $CPCT$).
Now join $AD$.
In $△$s $ADP$ and $ADQ$,
$AD=AD$ (common)
$DP=DQ$ (Proved above)
$\angle P=\angle Q={90}^o$(Given)
Thus, $ADP\cong ADQ$ ( By $SSA$ congruency).
Hence, $AP=AQ$.

That is, the given statement is true.

Therefore, option $A$ is correct.