Question

State true or false

$ABCD$ is a square. $A$ is joined to a point $P$ on $BC$ and $D$ is joined to a point $Q$ on $AB$, If $AP=DQ$, then $AP$ and $DQ$ are perpendicular to each other.

• A. True
• B. False

Answer 1

The correct option is A True

Given ABCD is a square & DQ=AP.

In $△$ ADQ & $△$ BAP,

AD=BA ( sides of square ABCD )

DQ=AP ( given )

$\angle ABP=\angle DAQ={90}^o$

$△$ ADQ$\cong △$ BAP( By RHS rule of congruency )

Therefore $\angle ADQ=\angle BAP$ -----( 1)

Now $\angle BAD=\angle DAP+\angle BAP$

$\Rightarrow \angle BAD=\angle DAP+\angle ADQ$ ---{ from 1 }

Since $\angle BAD={90}^o$ so $\angle DAP+\angle ADQ={90}^o$

$\Rightarrow \angle DAO+\angle ADO={90}^o$

In $△$ AOD, $\angle AOD+\angle ADO+\angle DAO={180}^o$

$\Rightarrow \angle AOD+{90}^o={180}^o$

$\Rightarrow \angle AOD={90}^O$

Hence proved that AP$\perp$ DQ.