SOLUTION: if the perimeter of a rectangle is 20 feet and the diagonal is 2 sqrt13 feet, then what are the length and width?

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Question 173514: if the perimeter of a rectangle is 20 feet and the diagonal is 2 sqrt13 feet, then what are the length and width?
Found 2 solutions by solver91311, Mathtut:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
I just did this one a couple of hours ago. See the solution to problem 174453.

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
P=2(L+W)...where L and W are length and width respectively
:
we are given the diagonal the hypothenuse of a right triangle where L and W are the legs
:
%282sqrt%2813%29%29%5E2=L%5E2%2BW%5E2...eq 1
20=2L%2B2W................eq 2
:
lets re write eq 1 for L. 2L=20-2W---->L=(20-2W)/2-->L=10-W
:
now lets plug in this L value into eq 2 and solve for W
:
4%2813%29=%2810-W%29%2810-W%29%2BW%5E2
:
52=100-20W%2BW%5E2%2BW%5E2multiplying out
:
2W%5E2-20W%2B48=0combining like terms on left side
:
W%5E2-10W%2B24=0dividing by 2
:
system%28W=4%2CW=6%29
:
W can be either 4 or 6 but when
:
W=4 then L=6 and when W=6 the L=4....
:
:
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B24+=+0) has the following solutons:

x%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=%28-10%29%5E2-4%2A1%2A24=4.

Discriminant d=4 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+4+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+4+%29%29%2F2%5C1+=+6
x%5B2%5D+=+%28-%28-10%29-sqrt%28+4+%29%29%2F2%5C1+=+4

Quadratic expression 1x%5E2%2B-10x%2B24 can be factored:
1x%5E2%2B-10x%2B24+=+%28x-6%29%2A%28x-4%29
Again, the answer is: 6, 4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B24+%29