Question

The base of triangle is of length $b$, and the altitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is

• A. $\frac{bx}{h}(h-x)$
• B. $\frac{hx}{h}(b-x)$
• C. $\frac{bx}{h}(h-2x)$
• D. $x(b-x)$
• E. $x(h-x)$

Answer 1

The correct option is A $\frac{bx}{h}(h-x)$

$△AOC\sim △EGC$ [AA similarly]

$\therefore \frac{AO}{EG}=\frac{OC}{GC}\Rightarrow \frac{h}{x}=\frac{b/2}{y}$

$\Rightarrow y=\frac{bx}{2h}\Rightarrow 2y=\frac{bx}{h}$

$\therefore$Base length of rectangle$=b-2y$

$=b-\frac{bx}{h}$

$=b[1-\frac{x}{h}]$

$\therefore$Area of rectangle$=x\times b[1-\frac{x}{h}]=\frac{bx(h-x)}{h}$