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

  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.

not enough information

step-by-step explanation:

we cannot know this, because the other students could have handed out 10 flyers, or they could have handed out 100. one way, the mean would go up, and another way, the mean would go down.