In the figure, the length of side AB of square ABCD is √50, E is between B and H, and BE = 1. What is the area of the inner square EFGH?

2005 AMC 10A Problem 8 — Solution

2005 AMC 10A Problem 8

Below is the professionally curated solution for Problem 8 of the 2005 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10A solutions, or check the answer key.

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Concepts:square (geometry)Pythagorean Theorem congruence (geometry)

Difficulty rating: 1280

In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50},$ $E$ is between $B$ and $H,$ and $BE = 1.$ What is the area of the inner square $EFGH?$

$25$

$32$

$36$

$40$

$42$

Solution:

The triangles $ABH, BCE, CDF,$ and $DAG$ are congruent right triangles. In $\triangle BCE$ the hypotenuse is $BC = \sqrt{50}$ and $BE = 1,$ so $CE = \sqrt{50 - 1} = 7.$ Since $BH = CE = 7$ and $E$ lies on $BH$ with $BE = 1,$ the inner square's side is $EH = 7 - 1 = 6,$ giving area $6^2 = 36.$

Thus, the correct answer is C.

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