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

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what are the lengths of its other two sides?
asked Mar 18, 2014 in GEOMETRY by rockstar Apprentice

1 Answer

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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.

answered Mar 24, 2014 by rob Pupil

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