Question

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

$3$

$\sqrt{10}$

$2 + \sqrt{2}$

$2\sqrt{3}$

$4$

Answer: A

Concepts:area ratio rectangle linear equation

Difficulty rating: 1340

Solution:

Let each rectangle have shorter side $x$ and longer side $y.$ The outer square has side length $y + x$ and the inner square has side length $y - x.$

Since the area ratio is $4,$ the side ratio is $2,$ so $y + x = 2(y - x),$ which gives $y = 3x.$

The ratio of longer to shorter side is $\dfrac{y}{x} = 3.$

Thus, the correct answer is A.

2009 AMC 10A Problem 14

Answer: A

Difficulty rating: 1340

Solution:

Problem 14 in Other Years