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 $AB?$

$4 + 4\sqrt5$

$10\sqrt2$

$5 + 5\sqrt5$

$10\sqrt[4]{8}$

$20$

Answer: D

Concepts:similarity area ratio rectangle

Difficulty rating: 1730

Solution:

The areas of the eleven pieces sum to $200.$ Let the enclosing rectangle have height $h$ and width $w=AB.$ Similar figures have corresponding side lengths in the square-root ratio of their areas. From the diagram, the area- $32$ piece contributes its short side to the left edge, of length $h\sqrt{32/200}=\tfrac25h.$ Above it, the area- $36$ piece contributes its long side, of length $w\sqrt{36/200}=\tfrac{3\sqrt2}{10}w.$ These two segments make the full height, so $h=\tfrac25h+\tfrac{3\sqrt2}{10}w,$ which simplifies to $h=w/\sqrt2.$ Since $wh=200,$ we get $w^2/\sqrt2=200$ and $w=\sqrt{200\sqrt2}=10\sqrt[4]{8}.$ Therefore, the answer is D.

2024 AMC 10A Problem 16

Answer: D

Difficulty rating: 1730

Solution:

Problem 16 in Other Years