Question

In a right triangle $\Delta ABC$ with right angle at B, a perpendicular BD is drawn from B to the hypotenuse. If AC = 2AB, find the area of $\Delta ADB$.
Given: Area of $\Delta ABC$ = 5 sq units

• A. 0.5 sq. unit
• B. 1 sq. unit
• C. 1.25 sq. unit
• D. 5 sq. unit

Answer 1

The correct option is C 1.25 sq. unit


Given: $\frac{AB}{AC}=\frac{1}{2}$
Ar($\Delta$ ABC) = 5 sq.units
Consider $\Delta ADB$ and $\Delta ABC$,
$\angle BAD=\angle BAC$ ----[commom angle]
$\angle BDA=\angle ABC$ ----[90${}^{\circ }$]
By AA similarity criterion, $\Delta ADB\sim \Delta ABC$
$\Rightarrow$ $\frac{Ar\left(\Delta ADB\right)}{Ar\left(\Delta ABC\right)}={\left(\frac{AB}{AC}\right)}^2={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$
$\Rightarrow$$Ar\left(\Delta ADB\right)\times 4=Ar\left(\Delta ABC\right)\Rightarrow Ar\left(\Delta ADB\right)=\frac{5}{4}=1.25sq.units$
Hence, the area of $\Delta ADB$ is 1.25 sq.units.