Question

In a right angled triangle $\Delta$ABC, a perpendicular BD is drawn on to the largest side from the opposite vertex. Which of the following does NOT give the ratios of the areas of $\Delta$ABD and $\Delta$ BCD?

• A. ${\left(\frac{AB}{BC}\right)}^2$
• B. ${\left(\frac{AB}{AC}\right)}^2$
• C. ${\left(\frac{AD}{BD}\right)}^2$
• D. ${\left(\frac{BD}{DC}\right)}^2$

Answer 1

The correct option is B ${\left(\frac{AB}{AC}\right)}^2$


Consider $\Delta ABD$ and $\Delta BCD$:

Since in $\Delta BDC$, $\angle DBC={90}^{\circ }-\angle C=\angle A$ and in $\Delta ABC,\angle A={90}^{\circ }-\angle C$,
$\angle BAD=\angle DBC$
$\angle BDA=\angle BDC$ $\left[{90}^{\circ }\right]$

By AA similarity criterion, $\Delta ADB\sim \Delta BDC$

Hence,
$\frac{Ar\left(\Delta ABD\right)}{Ar\left(\Delta BDC\right)}={\left(\frac{AB}{BC}\right)}^2={\left(\frac{AD}{BD}\right)}^2={\left(\frac{BD}{DC}\right)}^2$

Option B is the odd one out among these options as can be seen from the above equation.