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

Question

GPT-4o mini · Answer

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:

$(longer leg)^{2} + (shorter leg)^{2} = (hypotenuse)^{2}$

Substituting the expressions we have:

$x^{2} + (x - 3)^{2} = (x + 3)^{2}$

Now, let's expand and simplify the equation:

Expand $(x - 3)^{2}$ : $(x - 3)^{2} = x^{2} - 6 x + 9$
Expand $(x + 3)^{2}$ : $(x + 3)^{2} = x^{2} + 6 x + 9$

Now substitute these expansions back into the equation:

$x^{2} + (x^{2} - 6 x + 9) = (x^{2} + 6 x + 9)$

Combine like terms on the left side:

$2 x^{2} - 6 x + 9 = x^{2} + 6 x + 9$

Now, move all terms to one side:

$2 x^{2} - 6 x + 9 - x^{2} - 6 x - 9 = 0$

This simplifies to:

$x^{2} - 12 x = 0$

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