The following Level 5 geometry problem baffled a number of my students:

In triangle PQR, the length of side QR is 13 and the length of side PR is 21. What is the greatest possible integer length of side PQ?
(A) 9
(B) 17
(C) 26
(D) 28
(E) 33

Several of my students tried to use the Pythagorean Theorem while others tried to use advanced trigonometry. But their time-consuming efforts proved to be in vain.

The fastest way to solve this problem is to apply the Triangle Inequality Rule also known as the Third Side Rule. The rule states that, “the length of any side of a triangle must be between the sum and the difference of the lengths of the other two sides.” Or stated more simply – LESS THAN THE SUM AND GREATER THAN THE DIFFERENCE.

Let’s apply the Third Side Rule to this problem. The sum of the two given sides is 34. You are asked for the greatest possible integer length of the third side PQ. Since the sum of the two given sides is 34, the greatest possible length of the third side would be 34 – 1 = 33, so (E).

OK quick: What if the problem has asked you for the smallest possible integer length of side PQ? Using the Third Side Rule the answer would be 9 since 21 – 13 = 8 and one more than 8 would be 9!

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