AMC 8 Daily Practice Round 7

Complete problem set with solutions and individual problem pages

Problem 14 Easy

The perimeter of a triangle is 60cm, and its shortest side is 13cm. If all three sides of the triangle are integers, what is the maximum possible length of the longest side in centimeters?

  • A.

    24

  • B.

    25

  • C.

    28

  • D.

    29

  • E.

    30

Answer:D

Assume the three sides of the triangle, arranged in ascending order, are 13, a, and b.

According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side, so: 13 + a > b.

Since the perimeter of the triangle is 60 cm, we have: 13 + a + b = 60, which simplifies to: a = 47 - b.

Substituting a = 47 - b into the inequality 13 + a > b, we get: 13 + (47 - b) > b, 60 - b > b, 60 > 2b, b < 30.

Since b is an integer, the maximum possible value of b is 29.

Thus, the longest side can be at most 29 centimeters.

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