SOLUTION: if the perimeter of a rectangle is 20 feet anf the diagonal is 2 square root 13 feet, then what are the length and width?

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Question 173453: if the perimeter of a rectangle is 20 feet anf the diagonal is 2 square root 13 feet, then what are the length and width?
Answer by solver91311(24713) About Me  (Show Source):
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The perimeter of a rectangle is given by P=2l%2B2w, so we know that 2l%2B2w=20.

Solving for l:

2l%2B2w=20

2l=20-2w

l=10-w

The measure of a diagonal of a rectangle in terms of the length and width is given by the Pythagorean Theorem:

d=sqrt%28l%5E2%2Bw%5E2%29 where d is the measure of the diagonal.

But we know that: sqrt%28l%5E2%2Bw%5E2%29=2sqrt%2813%29

Square both sides:

l%5E2%2Bw%5E2=4%2A13=52

Now substitute the expression for l in terms of w developed earlier:

%2810-w%29%5E2%2Bw%5E2=4%2A13=52

Expand, collect terms, and put into standard form for a quadratic:

100-20w%2Bw%5E2%2Bw%5E2=52

2w%5E2-20w%2B48=0

Divide through by 2

w%5E2-10w%2B24=0

Leaving you with a factorable quadratic to solve. Hint: -4%2A-6=24 and -4-6=-10

Since you are solving for the width, pick the smaller of the two roots as your width and calculate the length from l=10-w (which you will find is the other root of your quadratic)