A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #12 #. What is the area of the triangle's incircle?

Question

A triangle has vertices A, B, and C.  Vertex A has an angle of #pi/8 #, vertex B has an angle of #(pi)/12  #, and the triangle's area is #12 #.  What is the area of the triangle's incircle?

Wyatt Anderson · Accepted Answer

#color(indigo)("Area of inscribed circle " = A_i = pi r^2 = 2.9092#

#"Area of "Delta = A_t = (1/2) a b sin C = (1/2) b c sin A = (1/2) c a sin B#

#"Given " hat A = pi/12, hat B = pi/8, hat C = (19pi)/24, A_t = 12#

#a b = (2 A_t) / sin C = 24 / sin ((19pi)/24) = 39.42#

#b c = (2 A_t) / sin A = 24 / sin (pi/12) = 92.73#

#c a = (2 A_t) / sin B = 24 / sin (pi/8) = 62.72#

#a = (a b c) / (b c) = sqrt(39.42 * 92.73 * 62.72) / 92.73= 5.16#

#b = (a b c) / (c a) = sqrt(39.42 * 92.73 * 62.72) / 62.72 = 7.63#

#c = (a b c) / (a b) = sqrt(39.42 * 92.73 * 62.72) / 39.42 = 12.15#

#"Semi-perimeter " = s = (a + b + c) / 2 = 24.94 / 2 = 12.47#

#"Radius of inscribed circle " = r = A_t / s = 12 / 12.47 = 0.9623#

#"Area of inscribed circle " = A_i = pi r^2 = pi * 0.9623^2 = 2.9092#

Sophia Alfred · Answer

undefined To find the area of the triangle's incircle, we can use the formula $A = rs$, where $A$ is the area of the triangle, $r$ is the radius of the incircle, and $s$ is the semi-perimeter of the triangle.

First, we need to find the lengths of the sides of the triangle using the given angles and the area.

Let $a$, $b$, and $c$ be the lengths of the sides opposite vertices A, B, and C respectively.

Using the formula for the area of a triangle $A = \frac{1}{2}ab\sin(C)$, we have:

$$
12 = \frac{1}{2}ab\sin(\frac{\pi}{8}) = \frac{1}{2}bc\sin(\frac{\pi}{12}) = \frac{1}{2}ca\sin(\frac{5\pi}{24})
$$

Solving this system of equations will give us the lengths of the sides $a$, $b$, and $c$.

Once we have the lengths of the sides, we can find the semi-perimeter $s = \frac{a+b+c}{2}$.

Then, we need to find the radius of the incircle, which can be calculated using the formula $r = \frac{A}{s}$, where $A$ is the area of the triangle and $s$ is the semi-perimeter.

Finally, we can find the area of the incircle using the formula $A_{\text{incircle}} = \pi r^2$.