To solve the problem of finding the perimeter of an isosceles right triangle with a hypotenuse of 100 inches, we need to follow a series of steps:
1. Understand the properties of an isosceles right triangle: In an isosceles right triangle, the two legs are of equal length, and the hypotenuse is opposite the right angle.
2. Relationship between the legs and the hypotenuse: For an isosceles right triangle, the hypotenuse can be found using the relationship:
[tex]\[
\text{Hypotenuse} = \text{leg} \times \sqrt{2}
\][/tex]
where the "leg" refers to one of the two equal sides.
3. Rearrange the formula to find the length of the legs:
[tex]\[
\text{leg} = \frac{\text{Hypotenuse}}{\sqrt{2}}
\][/tex]
4. Substitute the given hypotenuse (100 inches) into the formula:
[tex]\[
\text{leg} = \frac{100}{\sqrt{2}} \approx 70.71067811865474 \text{ inches}
\][/tex]
5. Calculate the perimeter: The perimeter of the triangle is the sum of all its sides, which includes two legs and the hypotenuse:
[tex]\[
\text{Perimeter} = 2 \times \text{leg} + \text{Hypotenuse}
\][/tex]
6. Substitute the known values into the perimeter formula:
[tex]\[
\text{Perimeter} = 2 \times 70.71067811865474 + 100 \approx 241.42135623730948 \text{ inches}
\][/tex]
Therefore, the perimeter of the isosceles right triangle with a hypotenuse of 100 inches is approximately 241.42135623730948 inches.
Now, let's compare this result with the provided answer choices.
None of the options directly match the calculated perimeter value of approximately 241.42135623730948 inches. However, we can reason through the answer choices:
- A) [tex]\(50 \sqrt{2}\)[/tex] does not match the calculated perimeter value.
- B) [tex]\(100 \sqrt{2}\)[/tex] does not match the calculated perimeter value.
- C) [tex]\(100 + 50 \sqrt{2}\)[/tex] does not match the calculated perimeter value.
- D) [tex]\(100 + 100 \sqrt{2}\)[/tex] does match the value provided as a numerical approximation.
The correct option that represents the perimeter is:
D) [tex]\(100 + 100 \sqrt{2}\)[/tex].
