How do I solve the following problem?
Instead of walking along the two sides of a rectangular field, Mario decided to cut across the diagonal. He saves a distance that is half of the long side of the field. Find the length of the long side of the field given that the short side is 123 feet.
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To answer this question, start by drawing a right triangle and label the short side, long side, and hypotenuse.
We start by noting that the short side is 123 feet. The long side is a bit more complicated. However, the distance is effectively given to us in the problem. If we let `l_h` be the length of the hypotenuse and `l_l` be the length of the long side, we get the following relation:
`l_h = 123+l_l - 1/2l_l = 123 + 1/2 l_l`
Notice here that we are simply "saving" the distance of one-half of the long side by going on the hypotenuse as opposed to traversing along the short and long legs!
We can now pair this relation with the pythagorean theorem in the following way:
`l_h = sqrt(123^2 + l_l^2)`
`123 + 1/2l_l = sqrt(123^2 + l_l^2)`
Now, we simplify by squaring both sides:
`(123+1/2l_l)^2 = 123^2 + l_l^2`
Now, simplify the squared binomial:
`123^2 + 123l_l + 1/4l_l^2 = 123^2 + l_l^2`
Subtract `123^2` and `l_l^2` from both sides:
`123l_l - 3/4 l_l^2 = 0`
The expresson on the left can be factored by removing `3/4l_l` in the following way. Keep in mind, we're going to be dividing 123 by `3/4 ` in the following operation to ensure complete factorization!
`3/4l_l(164 - l_l) = 0`
We can now divide both sides by `3/4l_l` because `l_l` is certainly not going to be zero!
`164 - l_l = 0`
Finally, add `l_l` to both sides to get our final result:
`164 = l_l`
I hope that helps!
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