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 square (geometry)system of equations

Difficulty rating: 1410

Solution:

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

Since the outer area is $4$ times the inner area, the side ratio is $\sqrt{4} = 2,$ so $x + y = 2(y - x).$

This gives $y = 3x,$ so the ratio of longer to shorter side is $3.$

Thus, the correct answer is A.

2009 AMC 12A Problem 8

Answer: A

Difficulty rating: 1410

Solution:

Problem 8 in Other Years