Question

A triangle has sides which are consecutive odd integers and an

angle of 120°. What is the perimeter of the triangle?

Answer 1

Perimeter = 15

Explanation:

We are told that the length of each of the 3 sides of the triangle are consecutive odd integers.

Thus means that we can write the 3 terms as;

(x - 2), x, (x + 2)

Thus, perimeter is;

P = (x - 2) + x + (x + 2)

P = 3x

If an angle is 120°, it means that the corresponding side that is directly opposite the angle is the largest side of the triangle.

Thus, this side is (x + 2)

Using cosine rule, we have;

(x + 2)² = x² + (x - 2)² - 2(x(x - 2))cos 120

Expanding, we have;

x² + 4x + 4 = x² + x² - 4x + 4 - (2x² + 4x)(-½)

This reduces to;

4x = x² - 4x + x² - 2x

2x² = 10x

2x = 10

x = 10/2

x = 5

Thus, other sides are;

(x - 2) = 5 - 2 = 3

And (x + 2) = 5 + 2 = 7

So,perimeter is;

P = 3x = 3 × 5 = 15