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


If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

$△ABC\sim △BDC$

and $△ABC\sim △ADB$

$\Rightarrow △BDC\sim △ADB$

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

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

Taking reciprocals, we get

$\frac{ar\left(△ADB\right)}{ar\left(△BDC\right)}={\left(\frac{AD}{BD}\right)}^2={\left(\frac{DB}{DC}\right)}^2={\left(\frac{AB}{BC}\right)}^2$

Therefore, from the given options, ${\left(\frac{AB}{AC}\right)}^2$ wouldn't give the ratio of the areas of $\Delta$ABD and $\Delta$ BCD.