Question

A right triangle has legs with lengths of x and x-7. If the length of the hypotenuse if 17 feet, what are the lengths of the legs of the right triangle?

Answer 1

Using the Pythagorean theorem and solving the resulting quadratic equation, the lengths of the legs of the right triangle with a hypotenuse of 17 feet are determined to be 15 feet and 8 feet.

Step-by-step explanation:

To find the lengths of the legs of a right triangle where the hypotenuse is 17 feet and the legs are of lengths x and x-7, we can use the Pythagorean theorem, which states that a² + b² = c², where a and b are the legs and c is the hypotenuse of the triangle.

Applying this theorem to our right triangle, we get x² + (x-7)² = 17². Expanding the equation, we have x² + x² - 14x + 49 = 289. Combining like terms, this simplifies to 2x² - 14x - 240 = 0.

Dividing the entire equation by 2, we get x² - 7x - 120 = 0. This is a quadratic equation which can be factored into (x-15)(x+8) = 0, yielding x = 15 or x = -8. Since a length cannot be negative, we discard the negative solution. Thus the lengths of the legs are 15 feet and 8 feet.