Question

Use the information, given in the adjoining figure, to show that AB = AC

Answer 1

Since PB, AD and QC are perpendiculars 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
BD = CD [Proved above]
AD = AD [common]
$\angle$ADB = $\angle$ADC = ${90}^{\circ }$
$\therefore \Delta$ABD $\cong \Delta$ACD [by SAS]
$\Rightarrow$ AB = AC [By C.P.C.T.C]