Question

Worth 20 points

Let n be a positive integer. In triangle ABC, AB = 3n, AC = 2n + 15, BC = n + 30, and angle A > angle B > angle C. How many possible values of n are there?

Answer 1

11

Explanation:

When the angle is greater, it's opposite length is greater

angle A > angle B > angle C

Implies,

BC > AC > AB

n + 30 > 2n + 15 > 3n

n + 30 > 2n + 15

n < 15 (1)

2n + 15 > 3m

n < 15 (2)

Also, the longest side of the triangle should be less than the sum of two shorter sides:

BC < AC + AB

n + 30 < (2n + 15) + 3n

n + 30 < 5n + 15

4n > 15

n > 3.75

n can be any integer betwee 4 and 14, inclusive.

Which are 11 possible values