Question

If the angles of a triangle are 45°, 45°, and 90°, show that the length of the hypotenuse is [√2] times as long as each leg.

Answer 1

The length of the hypotenuse of a triangle with angles of 45°, 45°, and 90° is [√2] times as long as each leg.

Step-by-step explanation:

The lengths of the legs of the triangle are both 45°. We can use the Pythagorean theorem, which states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b). In this case, a and b are both [√2] times as long as each leg, so:

• a = [√2] * length of each leg
• b = [√2] * length of each leg
• Substituting these values into the Pythagorean theorem, we get: c² = ([√2] * length of each leg)² + ([√2] * length of each leg)².
• Simplifying further, we get: c² = 2 * (length of each leg)² + 2 * (length of each leg)² = 4 * (length of each leg)².
• Taking the square root of both sides, we find: c = 2 * length of each leg = [√2] times as long as each leg.