Asked by M

A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sticks to make this happen?
Answered by Steve
clearly, when the longest side is the sum of the other two sides, the triangle has flattened into a line. So,

17-x = 8-x + 15-x
17-x = 23-2x
x = 6

cutting off 6" leaves the lengths of 2,9,11, a flat triangle.
Answered by Reiny
the sum of the two smaller sides must be greater than the largest side to have a triangle
so if we cut off x units from each stick

8-x + 15-x = 17-x
-x = -6
x = 6

so if we cut off 6 inches, our sticks will form a straight line, so
we can cut off <b>5 inches</b> from each and still get a triangle.
Answered by me
Our current triangle lengths are 8, 15, and 17. Let us say that x is the length of the piece that we cut from each of the three sticks. Then, our lengths will be 8 - x, 15 - x, and 17 - x. These lengths will no longer form a triangle when the two shorter lengths added together is shorter than or equal to the longest length. In other words, (8 - x) + (15 - x) ≤ (17 - x). Then, we have 23 - 2x ≤ 17 - x, so 6 ≤ x. Therefore, the length of the smallest piece that can be cut from each of the three sticks is *6 inches*.
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