SOLUTION: the length of one leg of a right triangle is 7 meters less than the length of the other leg. the length of the hypotenuse is 13 meters. find the lengths of the legs.

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Question 181935: the length of one leg of a right triangle is 7 meters less than the length of the other leg. the length of the hypotenuse is 13 meters. find the lengths of the legs.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

We basically have this triangle set up:





Since the legs are x and x-7 this means that a=x and b=x-7


Also, since the hypotenuse is 13, this means that c=13.


a%5E2%2Bb%5E2=c%5E2 Start with the Pythagorean theorem.


x%5E2%2B%28x-7%29%5E2=13%5E2 Plug in a=x, b=x-7, c=13


x%5E2%2B%28x-7%29%5E2=169 Square 13 to get 169.


x%5E2%2Bx%5E2-14x%2B49=169 FOIL


x%5E2%2Bx%5E2-14x%2B49-169=0 Subtract 169 from both sides.


2x%5E2-14x-120=0 Combine like terms.




Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-14, and c=-120


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-%28-14%29+%2B-+sqrt%28+%28-14%29%5E2-4%282%29%28-120%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-14, and c=-120


x+=+%2814+%2B-+sqrt%28+%28-14%29%5E2-4%282%29%28-120%29+%29%29%2F%282%282%29%29 Negate -14 to get 14.


x+=+%2814+%2B-+sqrt%28+196-4%282%29%28-120%29+%29%29%2F%282%282%29%29 Square -14 to get 196.


x+=+%2814+%2B-+sqrt%28+196--960+%29%29%2F%282%282%29%29 Multiply 4%282%29%28-120%29 to get -960


x+=+%2814+%2B-+sqrt%28+196%2B960+%29%29%2F%282%282%29%29 Rewrite sqrt%28196--960%29 as sqrt%28196%2B960%29


x+=+%2814+%2B-+sqrt%28+1156+%29%29%2F%282%282%29%29 Add 196 to 960 to get 1156


x+=+%2814+%2B-+sqrt%28+1156+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2814+%2B-+34%29%2F%284%29 Take the square root of 1156 to get 34.


x+=+%2814+%2B+34%29%2F%284%29 or x+=+%2814+-+34%29%2F%284%29 Break up the expression.


x+=+%2848%29%2F%284%29 or x+=++%28-20%29%2F%284%29 Combine like terms.


x+=+12 or x+=+-5 Simplify.


So the possible answers are x+=+12 or x+=+-5


However, a negative length is NOT possible. So x+=+-5 is NOT a solution


So the only answer is x+=+12





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Answer:


So the solution is x+=+12


This means that the other leg is 12-7=5


So the two legs are: 12 and 5 units