the shorter leg of a right triangle is 9 inches shorter than the longer leg. the hypotenuse is 9 inches longer than the longer leg. find the side lengths of the triangle

(L+9)^2=(L-9)^2 + L^2

EXPAND those out, then figure L using either factoring, or the quadratic equation.

To find the side lengths of the triangle, let's assign variables to the unknown values. Let's say that the longer leg is represented by 'x' inches.

According to the given information, the shorter leg is 9 inches shorter than the longer leg. So, the shorter leg can be represented as 'x - 9' inches.

Furthermore, the hypotenuse is 9 inches longer than the longer leg. So, the hypotenuse can be represented as 'x + 9' inches.

Now, we have the lengths of all three sides in terms of 'x':

Longer leg: x inches
Shorter leg: (x - 9) inches
Hypotenuse: (x + 9) inches

Since it is a right triangle, we can apply the Pythagorean Theorem, which states:
(c^2 = a^2 + b^2), where c is the hypotenuse, and a and b are the other two sides (legs).

Using the Pythagorean Theorem, we can write the equation for this triangle as:
(x + 9)^2 = x^2 + (x - 9)^2

Let's solve this equation to find the value of 'x':

(x + 9)^2 = x^2 + (x - 9)^2
x^2 + 18x + 81 = x^2 + x^2 - 18x + 81
x^2 + 18x + 81 = 2x^2 - 18x + 81

Simplifying further:
0 = 2x^2 - 36x

Dividing by 2:
0 = x^2 - 18x

Factoring out x:
x(x - 18) = 0

Setting each factor equal to zero:
x = 0 or x - 18 = 0

Ignoring the value x = 0 (as side lengths cannot be zero), we find that:
x - 18 = 0
x = 18

So, the longer leg is 18 inches.

Now, we can find the lengths of the other sides:

Shorter leg = x - 9 = 18 - 9 = 9 inches

Hypotenuse = x + 9 = 18 + 9 = 27 inches

Therefore, the side lengths of the right triangle are:
Longer leg: 18 inches
Shorter leg: 9 inches
Hypotenuse: 27 inches