Question bank / Lines, angles, and triangles

Math · Geometry and TrigonometryHard

Two sides of a triangle have lengths 77 and 1212. If the length of the third side is an integer, how many different lengths are possible for the third side?

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Answer: 13

By the triangle inequality, the third side xx must satisfy 12−7<x<12+712 - 7 < x < 12 + 7, or 5<x<195 < x < 19. The integers in this range are 66 through 1818, which is 18−6+1=1318 - 6 + 1 = 13 values.

Answering 1414 includes 55 or 1919, but a side equal to the difference or the sum of the other two sides would make a degenerate triangle.

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