2005 AMC 10A Problem 4

Problem 4 of 25EasierGeometry

A rectangle with a diagonal of length

x

is twice as long as it is wide. What is the area of the rectangle?

Show solution

Solution

Let the width be

w,

so the length is

2w.

Then

x^2 = w^2 + (2w)^2 = 5w^2,

giving

w^2 = \dfrac{x^2}{5}.

The area is

w \cdot 2w = 2w^2 = \dfrac{2}{5}x^2.

Thus, the correct answer is B.

Concepts: rectangle · Pythagorean Theorem · area

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.