SOLUTION: A rectangular parcel of land is 70 feet longer than it is wide. Each diagonal is 130 feet long. What are the dimensions of the parcel? I know you have to use the pythagorean the

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Question 183395: A rectangular parcel of land is 70 feet longer than it is wide. Each diagonal is 130 feet long. What are the dimensions of the parcel?
I know you have to use the pythagorean theorem but I have no clue what to do. Please help! :)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let L=length and W=width

Since the "land is 70 feet longer than it is wide", this tells us that L=W%2B70 (ie take the width and add 70 to get the length)


If you cut the rectangle in half along the diagonal, you'll find that the resulting pieces will be right triangles. So we can use Pythagorean's Theorem to find the legs (since we're given the hypotenuse)


a%5E2%2Bb%5E2=c%5E2 Start with Pythagorean's Theorem


L%5E2%2BW%5E2=130%5E2 Plug in a=L, b=W and c=130 (this is the given diagonal)


L%5E2%2BW%5E2=16900 Square 130 to get 16900


%28W%2B70%29%5E2%2BW%5E2=16900 Plug in L=W%2B70


W%5E2%2B140W%2B4900%2BW%5E2=16900 FOIL


W%5E2%2B140W%2B4900%2BW%5E2-16900=0 Subtract 16900 from both sides.


2W%5E2%2B140W-12000=0 Combine like terms.


Notice how the equation is now in the form ax%5E2%2Bbx%2Bc where a=2, b=140, and c=-12000


W+=+%28-140+%2B-+sqrt%28+%28140%29%5E2-4%2A2%2A-12000+%29%29%2F%282%2A2%29 Plug in a=2, b=140, and c=-12000



W+=+%28-140+%2B-+sqrt%28+19600-4%2A2%2A-12000+%29%29%2F%282%2A2%29 Square 140 to get 19600



W+=+%28-140+%2B-+sqrt%28+19600%2B96000+%29%29%2F%282%2A2%29 Multiply -4%2A-12000%2A2 to get 96000



W+=+%28-140+%2B-+sqrt%28+115600+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



W+=+%28-140+%2B-+340%29%2F%282%2A2%29 Simplify the square root


W+=+%28-140+%2B-+340%29%2F4 Multiply 2 and 2 to get 4



So now the expression breaks down into two parts


W+=+%28-140+%2B+340%29%2F4 or x+=+%28-140+-+340%29%2F4


Lets look at the first part:

W=%28-140+%2B+340%29%2F4


W=200%2F4 Add the terms in the numerator


W=50 Divide


So one possible answer is W=50



Now lets look at the second part:

W=%28-140+-+340%29%2F4


W=-480%2F4 Subtract the terms in the numerator


W=-120 Divide


So another possible answer is W=-120


However, it doesn't make sense to have a negative width. So we must ignore this possible solution



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So this means that the only answer is W=50 and that the width is 50 feet.


L=W%2B70 Go back to the first equation


L=50%2B70 Plug in W=50


L=120 Add


So the other answer is L=120 which means that the length is 120 feet.


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Answer:


So the length is 120 feet and the width is 50 feet.