SOLUTION: A right triangle has a height that is 6 inches longer than its shorter side. The hypotenuse of the triangle is 7 inches longer than the shorter side. How long is the hypotenuse of

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Question 955361: A right triangle has a height that is 6 inches longer than its shorter side. The hypotenuse of the triangle is 7 inches longer than the shorter side. How long is the hypotenuse of the triangle?
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
a=short side; b=a+6in; c=a+7in
a%5E2%2Bb%5E2=c%5E2 Substitute for b and c.
a%5E2%2B%28a%2B6%29%5E2=%28a%2B7%29%5E2
a%5E2%2Ba%5E2%2B12a%2B36=a%5E2%2B14a%2B49Subtract (a^2+14a+49) from each side.
a%5E2-2a-13=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aa%5E2%2Bba%2Bc=0 (in our case 1a%5E2%2B-2a%2B-13+=+0) has the following solutons:

a%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-13=56.

Discriminant d=56 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+56+%29%29%2F2%5Ca.

a%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+56+%29%29%2F2%5C1+=+4.74165738677394
a%5B2%5D+=+%28-%28-2%29-sqrt%28+56+%29%29%2F2%5C1+=+-2.74165738677394

Quadratic expression 1a%5E2%2B-2a%2B-13 can be factored:
1a%5E2%2B-2a%2B-13+=+1%28a-4.74165738677394%29%2A%28a--2.74165738677394%29
Again, the answer is: 4.74165738677394, -2.74165738677394. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-13+%29

a=4.74in
b=a+6=10.74in
c=a+7=11.74in ANSWER: The hypotenuse is 11.74 inches.
CHECK:
a%5E2%2Bb%5E2=c%5E2
4.74%5E2%2B10.74%5E2=11.74%5E2
22.47%2B115.35=137.82
137.82=137.82