SOLUTION: The hypotenuse of a right triangle is 12cm longer than the shortest side, and 6cm longer than the remaining side. Find the dimensions of the triangle

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Question 954664: The hypotenuse of a right triangle is 12cm longer than the shortest side, and 6cm longer than the remaining side. Find the dimensions of the triangle
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
c=hypotenuse; a=shortest leg=c-12cm; b=longer leg=c-6cm
a%5E2%2Bb%5E2=c%5E2 Substitute for a and b.
%28c-12cm%29%5E2%2B%28c-6cm%29%5E2=c%5E2
%28c%5E2-24c%2B144%29%2B%28c%5E2-12c%2B36%29=c%5E2
2c%5E2-36c%2B180=c%5E2 Subtract c%5E2 from each side.
c%5E2-36c%2B180=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ac%5E2%2Bbc%2Bc=0 (in our case 1c%5E2%2B-36c%2B180+=+0) has the following solutons:

c%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=%28-36%29%5E2-4%2A1%2A180=576.

Discriminant d=576 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--36%2B-sqrt%28+576+%29%29%2F2%5Ca.

c%5B1%5D+=+%28-%28-36%29%2Bsqrt%28+576+%29%29%2F2%5C1+=+30
c%5B2%5D+=+%28-%28-36%29-sqrt%28+576+%29%29%2F2%5C1+=+6

Quadratic expression 1c%5E2%2B-36c%2B180 can be factored:
1c%5E2%2B-36c%2B180+=+1%28c-30%29%2A%28c-6%29
Again, the answer is: 30, 6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-36%2Ax%2B180+%29

Possible solutions are 30 and 6 Since a=c-12, 6 doesn't work.
c=30 cm
ANSWER 1: The hypotenuse is 30 cm.
a=c-12cm=30-12cm=18cm
ANSWER 2: The shortest leg is 18 cm.
b=c-6cm=30-6cm=24cm
ANSWER 3: The longer leg is 24 cm.
CHECK:
a%5E2%2Bb%5E2=c%5E2
18%5E2cm%5E2%2B24%5E2cm%5E2=30%5E2cm%5E2
324cm%5E2%2B576cm%5E2=900cm%5E2
900cm%5E2=900cm%5E2