2006 AMC 10B Problem 10

Question

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?

43

44

45

46

47

Let the sides be

x,

3x,

and

15

The binding triangle inequality is

x+15 \gt 3x

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.

Problem 10 in other years