1991 AIME Problems/Problem 2
Contents
Problem
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.
Solution 1
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.
Solution 2
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,
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See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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