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:

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

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

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

$x(b-x)$

$x(h-x)$

Answer: A

Concepts:similarity rectangle area

Difficulty rating: 1280

Small Hint:

At height $x,$ use similarity to find the horizontal width of the triangle

Big Hint:

The available width is $b(1-\frac{x}{h})$ ; multiply it by the rectangle’s height

Solution:

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.

1960 AMC 12 Problem 37

Answer: A

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years