Official Guide Explanation:
Data Sufficiency #157

Background

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Solution and Metadata

Question: 157
Page: 287
Difficulty: 5 (Moderate)
Category 1: Geometry > Triangles > Special Tris
Category 2: Algebra > Linear Equations-Two Unk >

Explanation: If the hypotenuse of a right triangle is 10, we know that:

a² + b² = 10²

To find the perimeter, we'll need the specific values of a and b.

Statement (1) is sufficient. The legs of a right triangle (a and b) are the base and height, so we express the area as:

(1/2)ab = 25

ab = 50

It would take some involved algebra to find the precise values of a and b, but since we have two equations and two variables, we should be able to solve.

(Alert readers will note that, since the first equation involves exponents, these are not "linear" equations. If you do solve, you will find multiple possible answers. However, one of the two possible answers is negative, and in a geometry question, there can't be negative lengths. This is a typical "catch" in geometry Data Sufficiency questions.)

Statement (2) is also sufficient. It tells us that a = b. We can simplify the original equation and solve:

a² + a² = 100

2a² = 100

a² = 50

a = 5 rt[2]

Choice (D) is correct.

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