Question

In a right angled triangle ABC in which ∠A = ${90}^{\circ }$, If AD $\perp$ BC.

• A. ${AB}^2$ = BD × DC.
• B. ${AB}^2$ = BD × AD.
• C. ${AB}^2$ = BC × DC.
• D. ${AB}^2$ = BC × BD.

Answer 1

The correct option is D

${AB}^2$ = BC × BD.

Consider $\Delta$ BAC and $\Delta$ BDA,

$\angle$ ABC = $\angle$ DBA

$\angle$ BAC = $\angle$ BDA = ${90}^{\circ }$

$\angle$ BCA = $\angle$ BAD [Sum of angles of a triangle is constant]

$\therefore$ $△ABC$ is similar to $△DBA$ [By AAA similarity criterion]

$\frac{AB}{BD}$ = $\frac{BC}{AB}$ [Sides opposite to equal angles are proportional]

Hence, ${AB}^2$ = $BC\times BD$