The hypotenuse of a right triangle is 2 cm longer than the longer leg. The shorter leg is 7 cm shorter than the longer leg. Find the length of the longer leg of the triangle.

Answer 1

Let A be the shorter leg, and B be the longer leg, and H be the hypotenuse of the right triangle.

We have a right triangle, so we can apply the Pythagorean's theorem:

A² + B² = H²

Let's substitute H = B + 2 since the hypotenuse is said to be 2 cm longer than the longer leg,

and let's also substitute A = B - 7 since the shorter leg is said to be 7 cm shorter than the longer leg.

(B-7)² + B² = (B+2)²

Now we will go ahead and expand our binomial expressions

(B² -7B -7B + 49) + B² = (B² + 2B + 2B + 4)

Let's pass everything to the left-hand side.

(B² -7B -7B + 49) + B² - (B² + 2B + 2B + 4) = 0

Distribute the negative in front of the second parentheses, and remove the first parentheses.

B² -7B -7B + 49 + B² - B² - 2B - 2B - 4 = 0

Combine like terms

B² -18B + 45 = 0

Factor the left-hand side. You are looking for two numbers whose product is 45 and whose sum is -18. Those numbers are -15, and -3.

(B - 15)(B - 3) = 0

Using the Zero Product Property, we get:

B - 15 = 0 OR B -3 =0

Solving for B, you obtain

B = 15 or B = 3.

B, the longer leg cannot equal 3 cm because the shorter leg has to be 7 cm shorter than the longer leg, and a leg cannot be negative in length. Therefore, B, the longer leg equals 15 cm.