To determine the length of the hypotenuse in an isosceles right triangle where each leg measures 5 inches, we can follow these steps:
1. Understand the Properties of an Isosceles Right Triangle:
- This triangle has two equal legs and a right angle between them.
- The relationship between the legs and the hypotenuse in an isosceles right triangle can be expressed using the Pythagorean theorem.
2. Use the Pythagorean Theorem:
- For any right triangle, the square of the hypotenuse ([tex]\( c \)[/tex]) is equal to the sum of the squares of the other two sides ([tex]\( a \)[/tex] and [tex]\( b \)[/tex]):
[tex]\[
c^2 = a^2 + b^2
\][/tex]
- In an isosceles right triangle, [tex]\( a = b \)[/tex]. Let's set the leg length [tex]\( a = b = 5 \)[/tex] inches.
3. Plug in the Values:
[tex]\[
c^2 = 5^2 + 5^2
\][/tex]
This simplifies to:
[tex]\[
c^2 = 25 + 25
\][/tex]
[tex]\[
c^2 = 50
\][/tex]
4. Solve for the Hypotenuse:
[tex]\[
c = \sqrt{50}
\][/tex]
We can simplify [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
\][/tex]
5. Approximate the Hypotenuse:
- We know that [tex]\(\sqrt{2}\)[/tex] is approximately 1.414.
- Therefore:
[tex]\[
c = 5 \times 1.414 \approx 7.071
\][/tex]
6. Round to the Nearest Tenth:
- When rounding 7.071 to the nearest tenth, we get approximately 7.1.
So, the approximate length of the hypotenuse, rounded to the nearest tenth, is 7.1 inches. The correct answer is 7.1 inches.
