Question

The hypotenuse of a triangle is 3 inches longer than the longer leg. The shorter leg is 3 inches shorter than the longer leg. Find the lengths of the sides

Answer 1

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:

1. Expand \( (x - 3)^2 \):
\[
(x - 3)^2 = x^2 - 6x + 9
\]

2. 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:

1. \( x = 0 \) (not a valid side length)
2. \( x = 12 \)

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**