Question

All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $AB?$

$4 + 4\sqrt5$

$10\sqrt2$

$5 + 5\sqrt5$

$10\sqrt[4]{8}$

$20$

Solution:

Every piece is similar to the whole rectangle, so they all share one aspect ratio. The areas $1, 2, 4, 8, 16$ (and $9, 18$ ) come in factor-of- $2$ steps, and cutting a rectangle of aspect ratio $\sqrt2$ across its long side gives two similar copies of half the area. That pins the ratio at $\sqrt2.$ The total area is $36 + 16 + 8 + 18 + 1 + 9 + 4 + 2 + 32 + 25 + 49 = 200.$ The enclosing rectangle satisfies $AB \cdot \tfrac{AB}{\sqrt2} = 200,$ so $AB^2 = 200\sqrt2$ and $AB = \sqrt{200\sqrt2} = 10\sqrt[4]{8}.$ Therefore, the answer is D.

2024 AMC 10A Problem 16

Solution:

Problem 16 in Other Years