To find the hypotenuse of an isosceles right triangle when the legs are each of length 8, we can follow these steps:
1. Recall the properties of an isosceles right triangle: In such triangles, the two legs are of equal length, and the angles opposite these legs are each 45 degrees. The triangle is a special case of a right triangle.
2. Use the Pythagorean Theorem: According to this theorem, in any right triangle with legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and hypotenuse [tex]\(c\)[/tex], the relationship between these sides is given by:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
3. Substitute the leg lengths into the formula: Since both legs are equal in length and each leg is 8 units:
[tex]\[
8^2 + 8^2 = c^2
\][/tex]
4. Simplify the equation:
[tex]\[
64 + 64 = c^2
\][/tex]
[tex]\[
128 = c^2
\][/tex]
5. Solve for the hypotenuse [tex]\(c\)[/tex]:
[tex]\[
c = \sqrt{128}
\][/tex]
6. Simplify the square root:
[tex]\[
\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}
\][/tex]
7. Calculate the numerical value of [tex]\(8\sqrt{2}\)[/tex]:
[tex]\[
8\sqrt{2} \approx 8 \times 1.414213562 = 11.313708498984761
\][/tex]
So, the hypotenuse of the isosceles right triangle with legs of length 8 is approximately [tex]\(11.313708498984761\)[/tex].
