To solve the problem of finding the perimeter of an isosceles right triangle with a hypotenuse of length 58 inches, let's break down the solution step-by-step:
1. Identify properties of the triangle:
An isosceles right triangle has two equal legs, and a hypotenuse. For such triangles, if the hypotenuse is of length [tex]\( c \)[/tex], each leg is of length [tex]\( a \)[/tex].
2. Apply the Pythagorean theorem:
According to the Pythagorean theorem:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
Simplifying this, we get:
[tex]\[
2a^2 = c^2
\][/tex]
For our specific triangle, [tex]\( c = 58 \)[/tex]:
[tex]\[
2a^2 = 58^2
\][/tex]
3. Solve for [tex]\( a \)[/tex]:
[tex]\[
a^2 = \frac{58^2}{2}
\][/tex]
[tex]\[
a = \sqrt{\frac{58^2}{2}}
\][/tex]
Simplifying the expression inside the square root:
[tex]\[
a = \sqrt{\frac{3364}{2}}
\][/tex]
[tex]\[
a = \sqrt{1682}
\][/tex]
Now, calculating the actual value:
[tex]\[
a \approx 41.012193308819754
\][/tex]
4. Calculate the perimeter:
The perimeter [tex]\( P \)[/tex] of the triangle is the sum of all its sides. In this case, it includes the lengths of the two legs and the hypotenuse:
[tex]\[
P = a + a + c
\][/tex]
Substituting the calculated and given values:
[tex]\[
P \approx 41.012193308819754 + 41.012193308819754 + 58
\][/tex]
Simplifying, we get:
[tex]\[
P \approx 140.0243866176395
\][/tex]
Given the problem and multiple-choice answers, the correct answer among the provided options that matches this calculated perimeter is:
[tex]\[
C) 58 + 58 \sqrt{2}
\][/tex]
For verification, the approximate value of [tex]\( 58 \sqrt{2} \)[/tex] is:
[tex]\[
58 \cdot 1.414213562 = 82.425
\][/tex]
Adding 58 to this,
[tex]\[
58 + 82.425 \approx 140.425
\][/tex]
Thus, the closest option and correct answer is indeed:
[tex]\[
C) 58 + 58 \sqrt{2}
\][/tex]
