Question

the shorter leg of a right triangle is 1 centimeter less than the other leg. find the length of the two legs if the hypotenuse is 5 centimeters. the lengths of the two legs are cm. (use a comma to separate answers as needed.)

Explanation:

Step1: Set up variables

Let the longer leg be \(x\) cm. Then the shorter leg is \((x - 1)\) cm.

Step2: Apply Pythagorean theorem

By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(a=x - 1\), \(b = x\), \(c = 5\)), we have \((x-1)^{2}+x^{2}=5^{2}\).
Expand \((x - 1)^{2}\): \(x^{2}-2x + 1+x^{2}=25\).
Combine like terms: \(2x^{2}-2x+1 - 25=0\), so \(2x^{2}-2x - 24 = 0\).
Divide by 2: \(x^{2}-x - 12=0\).

Step3: Solve the quadratic equation

Factor \(x^{2}-x - 12=(x - 4)(x + 3)=0\).
Set each factor equal to zero: \(x-4=0\) gives \(x = 4\); \(x + 3=0\) gives \(x=-3\).
Since length cannot be negative, \(x = 4\).
The shorter leg is \(x-1=4 - 1=3\).

Answer:

\(3,4\)