SOLUTION: if one leg of a triangle is 14 meters shorter than the other leg, and the hypotenuse is 26 meters, find the length of the two legs.

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Question 156845: if one leg of a triangle is 14 meters shorter than the other leg, and the hypotenuse is 26 meters, find the length of the two legs.

Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!

Leg 1: x


Leg 2: x%2B14 (since one leg is 14 m shorter than the other, the other leg is 14 m longer than the first one)


Hypotenuse: 26



a%5E2%2Bb%5E2=c%5E2 Start with Pythagorean's Theorem


x%5E2%2B%28x%2B14%29%5E2=26%5E2 Plug in the given information


x%5E2%2B%28x%2B14%29%5E2=676 Square 26


x%5E2%2Bx%5E2%2B28x%2B196=676 FOIL


2x%5E2%2B28x%2B196=676 Combine like terms


2x%5E2%2B28x%2B196-676=0 Subtract 676 from both sides.


2x%5E2%2B28x-480=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=28, and c=-480


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%2828%29+%2B-+sqrt%28+%2828%29%5E2-4%282%29%28-480%29+%29%29%2F%282%282%29%29 Plug in a=2, b=28, and c=-480


x+=+%28-28+%2B-+sqrt%28+784-4%282%29%28-480%29+%29%29%2F%282%282%29%29 Square 28 to get 784.


x+=+%28-28+%2B-+sqrt%28+784--3840+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-480%29 to get -3840


x+=+%28-28+%2B-+sqrt%28+784%2B3840+%29%29%2F%282%282%29%29 Rewrite sqrt%28784--3840%29 as sqrt%28784%2B3840%29


x+=+%28-28+%2B-+sqrt%28+4624+%29%29%2F%282%282%29%29 Add 784 to 3840 to get 4624


x+=+%28-28+%2B-+sqrt%28+4624+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%28-28+%2B-+68%29%2F%284%29 Take the square root of 4624 to get 68.


x+=+%28-28+%2B+68%29%2F%284%29 or x+=+%28-28+-+68%29%2F%284%29 Break up the expression.


x+=+%2840%29%2F%284%29 or x+=++%28-96%29%2F%284%29 Combine like terms.


x+=+10 or x+=+-24 Simplify.


So the possible answers are x+=+10 or x+=+-24

However, a negative length isn't possible. So the only answer is x+=+10


So the first leg is 10 m, the second leg is 24 m (since 10%2B14=24), and the hypotenuse is 26 m