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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

The perimeter of an isosceles right triangle is $18 plus 18 StartRoot 2 EndRoot$ inches. What is the length, in inches, of the hypotenuse of this triangle?

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Explanation

Choice C is correct. The perimeter of a triangle is the sum of the lengths of its sides. Since the given triangle is an isosceles right triangle, the length of each leg is the same and the length of the hypotenuse is equal to $StartRoot 2 EndRoot$ times the length of a leg. Let $x$ represent the length, in inches, of a leg of this isosceles right triangle. Therefore, the perimeter, in inches, of the triangle is $x + x + x \sqrt{2}$ , or $2 x plus x StartRoot 2 EndRoot$ , which is equivalent to $x left parenthesis 2 plus StartRoot 2 EndRoot right parenthesis$ . It's given that the perimeter of this triangle is $18 plus 18 StartRoot 2 EndRoot$ inches. Thus, $x (2 + \sqrt{2}) = 18 + 18 \sqrt{2}$ . Dividing both sides of this equation by $2 plus StartRoot 2 EndRoot$ yields $x = \frac{18 + 18 \sqrt{2}}{2 + \sqrt{2}}$ . Multiplying the right-hand side of this equation by $\frac{2 - \sqrt{2}}{2 - \sqrt{2}}$ yields $x = \frac{36 + 36 \sqrt{2} - 18 \sqrt{2} - 36}{2}$ , or $x equals 9 StartRoot 2 EndRoot$ . It follows that the length, in inches, of a leg of this isosceles right triangle is $9 StartRoot 2 EndRoot$ . Therefore, the length, in inches, of the hypotenuse of this isosceles right triangle is $left parenthesis 9 StartRoot 2 EndRoot right parenthesis left parenthesis StartRoot 2 EndRoot right parenthesis$ , or $18$ .

Choice A is incorrect. If this were the length of the hypotenuse, the perimeter would be $9 plus 9 StartRoot 2 EndRoot$ inches.

Choice B is incorrect. This is the length, in inches, of a leg of this triangle, not the hypotenuse.

Choice D is incorrect. If this were the length of the hypotenuse, the perimeter would be $36 plus 18 StartRoot 2 EndRoot$ inches.