SOLUTION: i was never taught this and i don't know how to solve something like this, could you please help?? a right triangle has legs x+1. The hypotenuse of the triangle is 2x-1. What i

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Question 187798: i was never taught this and i don't know how to solve something like this, could you please help??
a right triangle has legs x+1. The hypotenuse of the triangle is 2x-1. What is the length of the hypotenuse?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
We basically have this triangle set up:




Remember, the Pythagorean Theorem is a%5E2%2Bb%5E2=c%5E2 where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.

Since both legs are x%2B1 this means that a=x%2B1 and b=x%2B1


Also, since the hypotenuse is 2x-1, this means that c=2x-1.



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


%28x%2B1%29%5E2%2B%28x%2B1%29%5E2=%282x-1%29%5E2 Plug in a=x%2B1, b=x%2B1, and c=2x-1


x%5E2%2B2x%2B1%2Bx%5E2%2B2x%2B1=4x%5E2-4x%2B1 FOIL


2x%5E2%2B4x%2B2=4x%5E2-4x%2B1 Combine like terms.


2x%5E2%2B4x%2B2-4x%5E2%2B4x-1=0 Get all terms to the left side.


-2x%5E2%2B8x%2B1=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=-2, b=8, and c=1


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-%288%29+%2B-+sqrt%28+%288%29%5E2-4%28-2%29%281%29+%29%29%2F%282%28-2%29%29 Plug in a=-2, b=8, and c=1


x+=+%28-8+%2B-+sqrt%28+64-4%28-2%29%281%29+%29%29%2F%282%28-2%29%29 Square 8 to get 64.


x+=+%28-8+%2B-+sqrt%28+64--8+%29%29%2F%282%28-2%29%29 Multiply 4%28-2%29%281%29 to get -8


x+=+%28-8+%2B-+sqrt%28+64%2B8+%29%29%2F%282%28-2%29%29 Rewrite sqrt%2864--8%29 as sqrt%2864%2B8%29


x+=+%28-8+%2B-+sqrt%28+72+%29%29%2F%282%28-2%29%29 Add 64 to 8 to get 72


x+=+%28-8+%2B-+sqrt%28+72+%29%29%2F%28-4%29 Multiply 2 and -2 to get -4.


x+=+%28-8+%2B-+6%2Asqrt%282%29%29%2F%28-4%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-8%2B6%2Asqrt%282%29%29%2F%28-4%29 or x+=+%28-8-6%2Asqrt%282%29%29%2F%28-4%29 Break up the expression.


x+=+%284%2B3%2Asqrt%282%29%29%2F%282%29 or x+=+%284%2B3%2Asqrt%282%29%29%2F%282%29 Reduce


So the answers are x+=+%284%2B3%2Asqrt%282%29%29%2F%282%29 or x+=+%284%2B3%2Asqrt%282%29%29%2F%282%29


which approximate to x=-0.121 or x=4.121


Since x=-0.121 generates a negative length, we'll ignore this answer.


So the only solution is x=4.121


This means that the two legs are

x%2B1=4.121%2B1=5.121


And the hypotenuse is

2%284.121%29-1=8.242-1=7.242

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Answer:


So the legs are approximately 5.121 units long and the hypotenuse is about 7.242 units long