A right triangle has legs of length 6 and 8. Find the length of the hypotenuse. Show the steps, then give brief study notes and one self-test question.

Leg $a = 6$
Leg $b = 8$

$6^2 + 8^2 = c^2$

$36 + 64 = c^2$

$100 = c^2$

$c = \sqrt{100}$
$c = 10$

The length of the hypotenuse is 10.

Brief Study Notes:

• The Pythagorean theorem ($a^2 + b^2 = c^2$) is fundamental for solving problems involving right triangles.
• Remember that $c$ always represents the hypotenuse, which is the longest side and is opposite the right angle.
• Pythagorean triples are sets of three positive integers $a, b, c$ that satisfy $a^2 + b^2 = c^2$ (e.g., 3-4-5, 5-12-13, 8-15-17). Recognizing these can speed up calculations. The triangle in this problem (6-8-10) is a multiple of the 3-4-5 triple.

Self-Test Question:

A right triangle has legs of length 5 and 12. What is the length of its hypotenuse?