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.

To find the length of the hypotenuse of a right triangle, we use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the legs, and c is the length of the hypotenuse. Steps: 1. Identify the given values: Leg a = 6 Leg b = 8 2. Substitute the values into the Pythagorean theorem: 6^2 + 8^2 = c^2 3. Calculate the squares of the legs: 36 + 64 = c^2 4. Add the squared values: 100 = c^2 5. Take the square root of both sides to find c: 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?