Let's denote the lengths of the sides of the triangle as follows:
- Let \( x \) be the length of the longer leg.
- Then, the hypotenuse can be expressed as \( x + 3 \) (3 inches longer than the longer leg).
- The shorter leg can be expressed as \( x - 3 \) (3 inches shorter than the longer leg).
According to the Pythagorean theorem, we have:
\[ (\text{longer leg})^2 + (\text{shorter leg})^2 = (\text{hypotenuse})^2 \]
Substituting the expressions we have:
\[ x^2 + (x - 3)^2 = (x + 3)^2 \]
Now, let's expand and simplify the equation:
-
Expand \( (x - 3)^2 \): \[ (x - 3)^2 = x^2 - 6x + 9 \]
-
Expand \( (x + 3)^2 \): \[ (x + 3)^2 = x^2 + 6x + 9 \]
Now substitute these expansions back into the equation:
\[ x^2 + (x^2 - 6x + 9) = (x^2 + 6x + 9) \]
Combine like terms on the left side:
\[ 2x^2 - 6x + 9 = x^2 + 6x + 9 \]
Now, move all terms to one side:
\[ 2x^2 - 6x + 9 - x^2 - 6x - 9 = 0 \]
This simplifies to:
\[ x^2 - 12x = 0 \]
Factoring out \( x \):
\[ x(x - 12) = 0 \]
This gives us two solutions:
- \( x = 0 \) (not a valid side length)
- \( x = 12 \)
Now that we have \( x \), we can find the lengths of the sides:
- Longer leg: \( x = 12 \) inches
- Shorter leg: \( x - 3 = 12 - 3 = 9 \) inches
- Hypotenuse: \( x + 3 = 12 + 3 = 15 \) inches
Thus, the lengths of the sides of the triangle are:
- Longer leg: 12 inches
- Shorter leg: 9 inches
- Hypotenuse: 15 inches