Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/4` and the angle between sides B and C is `pi/12`. If side B has a length of 8, what is the area of the triangle?

Answer 1

6.762 (6.762395692966 to be precise)

Explanation:

Answer can be verified here

Please use the above link to refer the image.

Now that we have the triangle (blue coloured in the above link), let's begin.

First drop a perpendicular from vertex A to the side BC, let it meet BC at the point D.

Now, assume the length BD = `x`
Thus, CD = `8-x`
(since BC = `8`)

using trigonometry, we have:

from triangle ADB, AD= `x cos(45º)` (because angle B=45º)
from triangle CDB, AD=`(8-x)cos(15º)` (because angle C=`π/12`=15º)

hence we have,
`x cos(45º)=(8-x)cos(15º)`
solve this linear equation in x to get the value of x as 2.4136.

hence AD = `x cos(45º) = 1.7067`

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Now, looking again at the triangle ABC whose area is to be found:
We have the height (AD=1.7067), we have the base (BC=8).

Thus, area of the triangle = `1/2 (base)(height) = 1/2 (1.7067)(8)`=6.762