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

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Answer 1 · 2016-11-17T21:14:39

The largest angle is $132˚$ .

Set up a systems of equations.

${(2x + 5 + d = 8x), (8x + d = 11x + 1):}$

$d = 6x - 5$

$->8x + 6x - 5 = 11x + 1$

$3x = 6$

$x = 2$

We can now find the sides of the triangle.

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

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.

$cosA = (b^2 + c^2 - a^2)/(2bc)$

$cosA = (16^2 + 9^2 - 23^2)/(2 xx 16 xx 9)$

$A = 132˚$

Hence, the largest angle is $132˚$ .

Hopefully this helps!

The sides of a triangle form three consecutive terms in an arithmetic sequence, with sides of length 2x + 52x+52x + 5, 8x8x8x, 11x + 111x+111x + 1. Determine the measure of the smallest angle within the triangle?