The hypotenuse of a right triangle is 65 inches long. One leg is 3 inches shorter than the other. Find the lengths of the legs of the triangle.

Question

The hypotenuse of a right triangle is 65 inches long. One leg is 3 inches shorter than the other. Find the lengths of the legs of the triangle.

Determine what quantities you are trying to find by defining your variables.

Formulate an equation that models the verbal conditions of the problem

Solve the equation.

Check your answer.

Answer 1

Draw a right triangle with 65 as the length of the hypotenuse.

One leg is 3 inches shorter than the other, so you could label their lengths x and x-3, where x stands for the length of the longer leg.

But addition is simpler than subtraction, so please label the lengths of the legs as x and x+3, where x stands for the length of the SHORTER leg.

The Pythagorean theorem tells us

hypotenuse² = (one leg)² + (the other leg)²

So

65² = x² + (x+3)²

4225 = x² + x² + 6x + 9

Let's combine the two x² terms:

4225 = 2x² + 6x + 9

Subtract 4225 from each side to get a zero all by itself on one side:

0 = 2x² + 6x - 4216

All the numbers are even, so we can divide both sides by 2 to make all the numbers smaller:

0 = x² + 3x - 2108

You can now find x using the quadratic formula with a=1, b=3, c=2108.

The two answers you get should be (1/2)(-3±√8441).

The length of a side of this triangle is positive, so the answer is the positive one: (1/2)(-3+√8441)

So the shorter leg of the triangle is of length (1/2)(-3+√8441)

and the longer leg is of length (1/2)(-3+√8441) + 3

Answer 2

65^2 = x^2 +(x-3)^2

4225 = 2x^2 -6x +9

2x^2 -6x -4216 =0

x^2 -3x -2108 = 0

complete the square and ignore the negative square root

x^2 -3x +(3/2)^2 = 2108+ 2.25

(x-1.5)^2 = 2110.25

x-1.5 = sqr2110.25

x = 1.5+ sqr2110.25

x = about 45.937 inches = Longer Leg

x-3 = about 42.937 inches = Shorter Leg

or about 46 and 43 inches for the 2 legs

Answer 3

Let x = length of short leg. Then, x + 3 = length of long leg

By the Pythagorean Theorem, x² + (x + 3)² = 65²

x² + x² + 6x + 9 = 4225

2x² + 6x - 4216 = 0

x² + 3x - 2108 = 0

Use the quadratic formula to solve for x.