Question

In a right angle triangle $△$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 $△$ABD and $△$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 $△ABD$ and $△BCD:$
$\angle BAD=\angle DBC$ [Since, in $△BDC,\angle DBC={90}^{\circ }-\angle C=\angle A$ and in $△ABC,\angle A={90}^{\circ }-\angle C$]
$\angle BDA=\angle BDC$ [${90}^{\circ }$]
By AA similarity criterion, $△ADB\sim △BDC$
Hence,

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

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