The hypotenuse of a right triangle is 9 feet more than the shorter leg and the longer leg is 15 feet. How do you find the length of the hypotenuse and shorter leg?

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Answer 1 · 2018-04-20T11:23:57

$hypotenuse = 17$ $color(blue)("hypotenuse"=17)$

$short leg = 8$ $color(blue)("short leg" = 8)$

Let $x$ $bbx$ be the length of the hypotenuse.

The shorter leg is 9 feet less than the hypotenuse, so the length of the shorter leg is:

$x - 9$ $x-9$

The longer leg is 15 feet.

By Pythagoras' theorem the square on the hypotenuse is equal to the sum of the squares of the other two sides:

$x^{2} = 15^{2} + {(x - 9)}^{2}$ $x^2=15^2+(x-9)^2$

So we need to solve this equation for $x$ $x$ :

$x^{2} = 15^{2} + {(x - 9)}^{2}$ $x^2=15^2+(x-9)^2$

Expand the bracket:

$x^{2} = 15^{2} + x^{2} - 18 x + 81$ $x^2=15^2+x^2-18x+81$

Simplify:

$306 - 18 x = 0$ $306-18x=0$

$x = \frac{306}{18} = 17$ $x=306/18=17$

The hypotenuse is $17$ $17$ feet long.

The shorter leg is:

$x - 9$ $x-9$

$17 - 9 = 8$ $17-9=8$ feet long.