Problem of the Day: 3/4/13

Two sides of a triangle measure 10 cm and 16 cm. What is the sum of the possible values of the other side?

Solution to yesterday’s problem:

Since the triangle has all integer lengths, it must be a Pythagorean triple. The side of the triangle that measures 24 inches could be the short side or the long side of the triangle. Let’s cover the cases where this side is the short side.

Pythagorean triples that could be candidates for this triangle are 3-4-5, 5-12-13, and 8-15-17. For 3-4-5, we know that if the short side is 8 times 3, then the long side would be 8 times 4, or 32. The triangle could not be 5-12-13 because 24 does not divide into 5. For 8-15-17, if the short side is 3 times 8, then the long side would be 3 times 25, or 45.

Moving on to the cases where the long side is 24 inches. Candidates for the triangle are again 3-4-5, 5-12-13, and 8-15-17. For 3-4-5, if the long side is 6 times 4, then the short side is 6 times 3, or 18. For 5-12-13, if the long side is 2 times 12, then the short side is 2 times 5, or 10.

Adding up all these values gives a final answer of 105.

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