A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(pi)/2 #, and the triangle's area is #2 #. What is the area of the triangle's incircle?
2 Answers
The area of the triangle's incircle is
Explanation:
Let
The simplest formula to find the inradius is
where
See this answer of mine for a generalised proof of this. As all triangles are tangential, the formula applies here aswell.
The angle of vertex
Our triangle is visualised below.
As it is a right triangle, its area will be equal to
Using the trigonometric functions, we see that
Using the sum and difference formulas, we find out that
Hence
Then,
Explanation:
Given that in
from sine in
Area of
Now, the in-radius (
Hence, the area of inscribed circle of