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


$Consider\Delta ABDand\Delta BCD:\angle BAD=\angle DBC(Sincein\Delta BDC,\angle DBC={90}^{\circ }-\angle C=\angle AandIn\Delta ABC\angle A={90}^{\circ }-\angle C)\angle BDA=\angle BDC------\left[{90}^{\circ }\right]\text{By AA similarity criterion,}\Delta ADB\sim \Delta BDCHence,\frac{Ar\left(\Delta ABD\right)}{Ar\left(\Delta BDC\right)}={\left(\frac{AB}{BC}\right)}^2={\left(\frac{AD}{BD}\right)}^2={\left(\frac{BD}{DC}\right)}^2$
Option B is the odd one out among these options as can be seen from the above equation.