A triangle has sides A, B, and C. The angle between sides A and B is (pi)/4. If side C has a length of 12 and the angle between sides B and C is ( 3 pi)/8, what are the lengths of sides A and B?

1 Answer
SCooke ยท sankarankalyanam
Mar 18, 2018

color(indigo)("Length of sides " A = B = 15.68

Explanation:

Given two angles, the third one in a triangle is fixed. In this case it is 3pi/8 (and we have an isosceles triangle!). The shortest side length will be opposite the smallest angle, which is 2pi/8 in this case. We know that the side of length 12 is opposite the 2pi/8 corner.

We now have three angles and a side, and can calculate the other sides using the Law of Sines, and then calculate the height for the area.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-sines
https://www.mathsisfun.com/algebra/trig-solving-asa-triangles.html

![http://www.dummies.com/education/math/trigonometry/laws-of-sines-and-cosines/](useruploads.socratic.org)

a/(sin((3pi)/8)) = c/sin C = 12/(sin((2pi)/8))
b/(sin((3pi)/8)) = c/sin C = 12/(sin((2pi)/8))

a xx sin((2pi)/8) = 12 xx sin((3pi)/8)

b xx sin((2pi)/8) = 12 xx sin((3pi)/8)

Since hat A = hat B, " it's an isosceles triangle"

a xx 0.707 = 12 xx 0.924 ; color(brown)(a = 15.68 = b

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