Question

In a right angle 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 $\Delta BCD$:
$\angle BAD=\angle DBC$
(Since in $\Delta BDC,\angle DBC={90}^{\circ }-\angle C=\angle A$
and in $\Delta ABC,\angle A={90}^{\circ }-\angle C)$

$\angle BDA=\angle BDC$ ....... [Both are 90${}^{\circ }$ angles]
$\therefore$ By AA similarity criterion, $\Delta ADB\sim \Delta BDC$
Hence,
$\frac{Area\left(\Delta ADB\right)}{Area\left(\Delta BDC\right)}={\left(\frac{AB}{BC}\right)}^2={\left(\frac{AD}{BD}\right)}^2={\left(\frac{BD}{DC}\right)}^2$
Option B is the odd one out among these options as can be seen from the above equation.