Question

In the given figure, AP = AQ and the right agles are shown. Which of the following statements can certainly be stated?

• A. AP = PB
• B. AB = AC
• C. AQ = AD
• D. None of these

Answer 1

The correct option is B

AB = AC

Since PB, AD and QC are perpendicular to the same line BC, they are parallel to each other i.e. PB || AD || QC.
Since, PB || AD || QC and PQ is a transversal making equal intercepts i.e. PA = AQ; therefore the other transversal BC will also make equal intercepts i.e. BD = CD.
Now in $\Delta ABD$ and $\Delta ACD$.
(i) BD = CD [Proved above]
(ii) AD = AD [Common]
(iii) $\angle ADB=\angle ADC={90}^{\circ }$ [As, AD $\perp$ BC]
$\therefore$ $\Delta ABD=\Delta ACD$ [By SAS]
$\Rightarrow$ AB = AC [By C.P.C.T]