1960 AMC 12 Problem 37

Question

The base of a 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:

\dfrac{bx}{h}(h-x)

\dfrac{hx}{b}(b-x)

\dfrac{bx}{h}(h-2x)

x(b-x)

x(h-x)

At height

x,

use similarity to find the horizontal width of the triangle

The available width is

b(1-\frac{x}{h})

; multiply it by the rectangle’s height

The cross-section parallel to the base at height

x

has width

b\left(1-\frac{x}{h}\right)=\frac{b(h-x)}{h}

by similarity. Multiplying by the rectangle’s height

x

gives

\frac{bx}{h}(h-x).

Thus, the correct answer is A.

Problem 37 in other years