Question

the shorter leg of a right triangle is 14 feet less than the other leg. find the length of the two legs if the hypotenuse is 26 feet. the length of the shorter leg of the right triangle is and the length of the other leg of the right triangle is

Explanation:

Step1: Define variables

Let the length of the longer leg be \(x\) feet. Then the length of the shorter leg is \((x - 14)\) feet.

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse). So we have \((x - 14)^{2}+x^{2}=26^{2}\).
Expand \((x - 14)^{2}\): \((x - 14)^{2}=x^{2}-28x + 196\).
The equation becomes \(x^{2}-28x + 196+x^{2}=676\).
Combine like - terms: \(2x^{2}-28x+196 - 676=0\), \(2x^{2}-28x - 480 = 0\).
Divide the entire equation by \(2\): \(x^{2}-14x - 240=0\).

Step3: Solve the quadratic equation

Factor the quadratic equation \(x^{2}-14x - 240=(x - 24)(x + 10)=0\).
Set each factor equal to zero: \(x-24 = 0\) or \(x + 10=0\).
We get \(x = 24\) or \(x=-10\). Since length cannot be negative, we discard \(x=-10\).

Step4: Find the lengths of the legs

If \(x = 24\) (longer leg), then the shorter leg is \(x-14=24 - 14=10\).

Answer:

The length of the shorter leg of the right triangle is \(10\) feet and the length of the other leg of the right triangle is \(24\) feet.