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?

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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!