A triangle has sides A, B, and C. The angle between sides A and B is #(5pi)/6#. If side C has a length of #2 # and the angle between sides B and C is #pi/12#, what are the lengths of sides A and B?

1 Answer
jim p. · Shwetank Mauria
Oct 24, 2017

A = B = 1.035

Explanation:

This one is deceptively easy, I think.

You know that the interior angles of a triangle add up to #pi# radians (or 180 degrees). You are given 2 of the angles, so you can calculate the angle between sides A and C = #pi - (pi/12 + (5pi)/6)#

...this comes out to #pi/12#. This is the same as the angle between sides B & C.

So you know you have an isosceles triangle, and therefore sides A and B are equal.

From your equilateral triangle, you can create two congruent right triangles, with hypotenuse A (or B), and base of length #C_1# and #C_2#. (#C_1 + C_2 = 2#, and since the two right triangles are equal, #C_1 = C_2#, so #C_1 = C_2 = 1#)

So now, using just a little trig, we know: #C_1/A = cos(pi/12)#
...since #C_1 = 1#, we have #1/A = cos(pi/12)#
Therefore, #A = 1/cos(pi/12)#

#A = 1.035# (rounding)

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