1991 AIME Problems/Problem 2
Rectangle has sides of length 4 and of length 3. Divide into 168 congruent segments with points , and divide into 168 congruent segments with points . For , draw the segments . Repeat this construction on the sides and , and then draw the diagonal . Find the sum of the lengths of the 335 parallel segments drawn.
The length of the diagonal is (a 3-4-5 right triangle). For each , is the hypotenuse of a right triangle with sides of . Thus, its length is . Let . We want to find since we are over counting the diagonal.
Using the above diagram, we have that and each one of these is a dilated 3-4-5 right triangle (This is true since is a 3-4-5 right triangle). Now, for all , we have that is the hypotenuse for the triangle . Therefore we want to know the sum of the lengths of all .This is given by the following:
Then by the summation formula for the sum of the terms of an arithmetic series,
First, count the diagonal which has length . For the rest of the segments, think about pairing them up so that each pair makes . For example, the parallel lines closest to the diagonal would have length while the parallel line closest to the corner of the rectangle would have length by similar triangles. If you add the two lengths together, it is There are pairs of these segments, for a total of ~justlearningmathog
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