a right triangle has an area of 84 square feet and a hypotenuse of 25 feet long

Question

what are the lengths of its other two sides?

Answer 1

The area of right triangle is 84 square feet and its hypotenuse is 25 feet.

The area of right triangle is A = 1/2 base * height.

Let x be the base (adjacent side) and y be the height (opposite side) of the right triangle.

To find the value of y, apply Pythagorean theorem : hypotenuse² = adjacent side² + opposite side²

(25)² = x² + y².

y² = 625 - x²

y = sqrt (625 - x²)

Area of right triangle : A = 1/2 base * height.

84 = 1/2 x  * sqrt (625 - x²)

168 = x  * sqrt (625 - x²)

square both sides.

28224 = x² * (625 - x²)

28224 = 625x² - x⁴.

 x⁴ - 625x² + 28224 = 0

x⁴ - 576x² - 49x²+ 28224 = 0

x²(x² - 576) - 49(x² - 576) = 0

(x² - 49)(x² - 576) = 0

(x² - 49) = 0 and (x² - 576) = 0

x² = 49 and x² = 576

x = ± 7 and x = ± 24.

Take only positive value of x, since the measurements never expressed with negative sign.

If x = 7 then y = sqrt (625 - x²) = sqrt (625 - 7²) = sqrt (625 - 49) = sqrt 576 = 24.

If x = 24 then y = sqrt (625 - x²) = sqrt (625 - 24²) = sqrt (625 - 576) = sqrt 49 = 7.

The remaining two sides of the triangle are 7 feet and 24 feet.