A triangle has sides A, B, and C. The angle between sides A and B is pi/3π3 and the angle between sides B and C is pi/12π12. If side B has a length of 1, what is the area of the triangle?

1 Answer
Roella W.
Jan 19, 2016

A~~ 0.116A≈0.116

Explanation:

Sketch
The area of a triangle A = 1/2*Base*heightA=12⋅Base⋅height
In this case the base B=1B=1
If we name the two portions of the base on either side of the perpendicular hh as xx and yy, then B = x+y = 1B=x+y=1

We then use the trigonometric relations
h/x = tan(pi/12)hx=tan(π12)
h/y =tan(pi/3)hy=tan(π3)

We now have three equations and three unknowns so we can solve to find hh
x = h/tan(pi/12)x=htan(π12)
y = h/tan(pi/3)y=htan(π3)
:.h/tan(pi/12) + h/tan(pi/3) = 1

h(tan(pi/3) + tan(pi/12))/(tan(pi/12)tan(pi/3)) = 1

:.h = (tan(pi/12)tan(pi/3))/(tan(pi/3) + tan(pi/12))

A = 1/2*1*(tan(pi/12)tan(pi/3))/(tan(pi/3) + tan(pi/12))

A~~( 0.268*1.732)/(2(1.732 + 0.268))

A~~ 0.464/4 ~~ 0.116

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