Question

One leg of a right triangle is twice the length of the other leg. The length of the hypotenuse is √45 centimeters. Let x represent the length of the shorter leg. Use the Pythagorean Theorem to write and solve an equation to find the length of the legs.

Answer 1

Let's use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, we are given that one leg (let's call it the shorter leg) is twice the length of the other leg. So, if we let x represent the length of the shorter leg, then the longer leg has a length of 2x.

We are also given that the length of the hypotenuse is √45 centimeters. We can simplify this by noticing that √45 = √(9 × 5) = √9 × √5 = 3√5. So the length of the hypotenuse is 3√5 centimeters.

Now we can write the Pythagorean Theorem equation:

x^2 + (2x)^2 = (3√5)^2

Simplifying, we get:

x^2 + 4x^2 = 45

5x^2 = 45

x^2 = 9

x = 3

So the shorter leg has a length of 3 centimeters, and the longer leg has a length of 2x = 2(3) = 6 centimeters.