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 ${\left(\frac{AB}{AC}\right)}^2$

In $△ABC$,
$\angle A={90}^{\circ }-\angle C$ ($\because$ $\angle B={90}^{\circ }$)

In $△ABD$ and $△BCD$,
$\angle DBC={90}^{\circ }-\angle C$
$\angle A=\angle DBC$
$\angle BDA=\angle BDC={90}^{\circ }$
$\therefore$ $△ABD\sim △BCD$ (By AA similarity)

$\Rightarrow$ $\frac{ar\left(△ABD\right)}{ar\left(△BCD\right)}=\frac{A{B}^2}{B{C}^2}=\frac{B{D}^2}{C{D}^2}=\frac{A{D}^2}{B{D}^2}$