Exponents and Polynomials
Using the Pythagorean Theorem and a quadratic equation to find side..

The shorter leg of a right triangle is $5 \mathrm{~m}$ shorter than the longer leg. The hypotenuse is $5 \mathrm{~m}$ longer than the longer leg. Find the side lengths of the triangle.
Length of the shorter leg: \m
Length of the longer leg: $\square \mathrm{m}$
Length of the hypotenuse: $\quad \square \mathrm{m}$

Step 1 ：Let's denote the length of the longer leg as \( x \).

Step 2 ：The shorter leg is \( x - 5 \) meters.

Step 3 ：The hypotenuse is \( x + 5 \) meters.

Step 4 ：Using the Pythagorean Theorem, we have \( (x - 5)^2 + x^2 = (x + 5)^2 \).

Step 5 ：Expanding and simplifying the equation, we get \( 2x^2 - 20x + 25 = x^2 + 10x + 25 \).

Step 6 ：Subtracting \( x^2 + 10x + 25 \) from both sides, we have \( x^2 - 30x = 0 \).

Step 7 ：Factoring out an \( x \), we get \( x(x - 30) = 0 \).

Step 8 ：Setting each factor equal to zero, we have \( x = 0 \) or \( x - 30 = 0 \).

Step 9 ：Since the length of a side cannot be zero, we disregard the solution \( x = 0 \).

Step 10 ：Solving for \( x - 30 = 0 \), we get \( x = 30 \).

Step 11 ：Therefore, the length of the longer leg is \( 30 \) meters.

Step 12 ：The length of the shorter leg is \( 30 - 5 = 25 \) meters.

Step 13 ：The length of the hypotenuse is \( 30 + 5 = 35 \) meters.