# The sides of a triangle form three consecutive terms in an arithmetic sequence, with sides of length 2x + 5, 8x, 11x + 1. Determine the measure of the smallest angle within the triangle?

Nov 17, 2016

The largest angle is 132˚.

#### Explanation:

Set up a systems of equations.

$\left\{\begin{matrix}2 x + 5 + d = 8 x \\ 8 x + d = 11 x + 1\end{matrix}\right.$

$d = 6 x - 5$

$\to 8 x + 6 x - 5 = 11 x + 1$

$3 x = 6$

$x = 2$

We can now find the sides of the triangle.

$2 \left(2\right) + 5 = 9$
$8 \left(2\right) = 16$
$11 \left(2\right) + 1 = 23$

The largest angle will be opposite the largest side.

Let's call the side that measures $23$ a, the side that measures $16$ b, and the side that measures $9$ c.

$\cos A = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c}$

$\cos A = \frac{{16}^{2} + {9}^{2} - {23}^{2}}{2 \times 16 \times 9}$

A = 132˚

Hence, the largest angle is 132˚.

Hopefully this helps!