Question

The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet? A. 26B. 32C. 40D. 64E. 80

Answer 1

Answer:

  A.  26 ft

Step-by-step explanation:

You might recognize that this triangle is a 5-12-13 triangle scaled up by a factor of 2. If so, you know immediately, that the hypotenuse is 2×13 = 26 feet in length.

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If you're not so lucky, you can write equations involving the lengths of the legs of the triangle. Let those be x and y, with x being the shorter leg. Then the Pythagorean relation tells you ...

  x² + y² = (x+16)²

and the area relation tells you ...

  120 = (1/2)xy

Using the second equation to write an expression for y, we have ...

  y = 240/x

Substituting this into the first equation gives ...

  x² + (240/x)² = (x+16)²

  (240/x)² = 32x +256 . . . . subtract x²

  1800 = x²(x +8) . . . . . . . . .multiply by x²/32

This is a cubic with one real root. It can be solved several ways. I like "graphing" as it is quick and easy. Trial and error also works. The factors of 1800 include 10²·18, so it is clear that x=10 is the solution.

The hypotenuse length is x+16, so is 26 feet.

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The attachment shows the graphical solution of the original two equations. This tells you the legs of the triangle are 10 and 24, so the hypotenuse will be 10+16 = 26. The 10-24-26 triangle is recognizable as a scaled version of the 5-12-13 triangle often used in math problems.