2006 AMC 10B Problem 10

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is

15.

What is the greatest possible perimeter of the triangle?

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Solution

Let the sides be

x,

3x,

and

15.

The triangle inequality

x+15 \gt 3x

gives

x \lt 7.5.

The largest integer is

x=7,

giving sides

7,

21,

15

and perimeter

7+21+15=43.

Thus, the correct answer is A.

Tagged: triangle inequality · perimeter · optimization