Question

one leg of a right triangle is 1 millimeter shorter than the longer leg and the hypotenuse is 1 millimeter longer than the longer leg. find the lengths of the sides of the triangle. the length of the longer leg is

Explanation:

Step1: Define variables

Let the length of the longer leg be \( x \) millimeters. Then the shorter leg is \( x - 1 \) millimeters, and the hypotenuse is \( x + 1 \) millimeters.

Step2: Apply Pythagorean theorem

For a right triangle, \( (\text{shorter leg})^2 + (\text{longer leg})^2 = (\text{hypotenuse})^2 \). Substituting the values, we get:
\[
(x - 1)^2 + x^2 = (x + 1)^2
\]

Step3: Expand the equations

Expand each term:
\[
x^2 - 2x + 1 + x^2 = x^2 + 2x + 1
\]

Step4: Simplify the equation

Combine like terms:
\[
2x^2 - 2x + 1 = x^2 + 2x + 1
\]
Subtract \( x^2 + 2x + 1 \) from both sides:
\[
2x^2 - 2x + 1 - (x^2 + 2x + 1) = 0
\]
\[
2x^2 - 2x + 1 - x^2 - 2x - 1 = 0
\]
\[
x^2 - 4x = 0
\]

Step5: Solve for \( x \)

Factor out \( x \):
\[
x(x - 4) = 0
\]
So, \( x = 0 \) or \( x - 4 = 0 \). Since length can't be 0, \( x = 4 \).

Step6: Find all side lengths

Longer leg: \( x = 4 \) mm.
Shorter leg: \( x - 1 = 4 - 1 = 3 \) mm.
Hypotenuse: \( x + 1 = 4 + 1 = 5 \) mm.

Answer:

The lengths of the sides are 3 millimeters, 4 millimeters, and 5 millimeters. The length of the longer leg is 4 millimeters.