Question

The altitude of a triangle is two-fifth the length of its corresponding base. If the altitude is increased by 4 cm and the base is decreased by 2 cm, then the area of the triangle remains the same. Find the base of the triangle in cm.

Answer 1

$\text{Let the base of the triangle be}xcm.$

$\text{Then, the corresponding altitude}\text{of the triangle would be}\frac{2x}{5}cm.$

$\text{If altitude is increased by 4 cm,}\text{the new altitude}=\frac{2x}{5}+4cm$

$\text{New base}=x-2cm$

$\text{Area of triangle}=\frac{1}{2}\times \text{base}\times \text{height}$

$\text{Since the area remains the same,}\frac{1}{2}\times x\times \frac{2x}{5}=\frac{1}{2}\times (\frac{2x}{5}+4)\times (x-2)$
$\Rightarrow \frac{x^2}{5}=\frac{x^2}{5}+2x-\frac{2x}{5}-4$
$\Rightarrow 2x-\frac{2x}{5}=4\Rightarrow \frac{10x-2x}{5}=4$
$\Rightarrow 8x=20$
$\Rightarrow x=\frac{20}{8}\Rightarrow x=2\frac{1}{2}\text{cm}$
$\text{Hence, base of the triangle is}2\frac{1}{2}cm\text{long.}$