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An isosceles right triangle has a hypotenuse of length $58$ inches. What is the perimeter, in inches, of this triangle?

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An isosceles right triangle has a hypotenuse of length $58$ inches. What is the perimeter, in inches, of this triangle?

","comment":{"@type":"Comment","text":"

Choice C is correct. Since the triangle is an isosceles right triangle, the two sides that form the right angle must be the same length. Let $x$ be the length, in inches, of each of those sides. The Pythagorean theorem states that in a right triangle, $a^{2}+b^{2}=c^{2}$, where $c$ is the length of the hypotenuse and $a$ and $b$ are the lengths of the other two sides. Substituting $x$ for $a$, $x$ for $b$, and $58$ for $c$ in this equation yields $\\bm{x}^{2}+x^{2}=58^{2}$, or $2x^{2}=58^{2}$. Dividing each side of this equation by $2$ yields $x^{2}=\\frac{58^{2}}{2}$, or $x^{2}=\\frac{2.58^{2}}{4}$. Taking the square root of each side of this equation yields two solutions: $x=\\frac{58\\sqrt{2}}{2}$ and $x=-\\frac{58\\sqrt{2}}{2}$. The value of $x$ must be positive because it represents a side length. Therefore, $x=\\frac{58\\sqrt{2}}{2}$, or $x=29\\sqrt{2}$. The perimeter, in inches, of the triangle is $58+x+x$, or $58+2x$. Substituting $29\\sqrt{2}$ for $x$ in this expression gives a perimeter, in inches, of $58+2\\left(29\\sqrt{2}\\right)$, or $58+58\\sqrt{2}$.

Choice A is incorrect. This is the length, in inches, of each of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.

Choice B is incorrect. This is the sum of the lengths, in inches, of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.

Choice D is incorrect and may result from conceptual or calculation errors.

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An isosceles right triangle has a hypotenuse of length $58$ inches. What is the perimeter, in inches, of this triangle?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$29\\sqrt{2}$

"},{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$58\\sqrt{2}$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$58+116\\sqrt{2}$

"}],"acceptedAnswer":{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$58+58\\sqrt{2}$

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