A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/8 #, and the triangle's area is #8 #. 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/12 #, vertex B has an angle of #(5pi)/8  #, and the triangle's area is #8 #.  What is the area of the triangle's incircle?

Savannah Aguilar · Accepted Answer

#color(magenta)("Area of inscribed circle " = A_i = pi r^2 ~~ 2.442#

#"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 = (5pi)/8, hat C = ((7pi)/24)#, A_t = 8#

#a b = (2 A_t) / sin C = 16 / sin ((7pi)/24) ~~ 20.1676#

#b c = (2 A_t) / sin A = 16 / sin (pi/12) = 61.8193#

#c a = (2 A_t) / sin B = 16 / sin ((5pi)/8) = 17.3183#

#a = (a b c) / (b c) = sqrt(20.1676 * 61.8193 * 17.3183) / 61.8193 = 2.3769#

#b = (a b c) / (c a) = sqrt(20.1676 * 61.8193 * 17.3183) / 17.3183 = 8.4847#

#c = (a b c) / (a b) = sqrt(2.3769 * 61.8193 * 17.3183) / 20.1676 = 7.286#

#"Semi-perimeter " = s = (a + b + c) / 2 = 18.1476 / 2 = 9.0738#

#"Radius of inscribed circle " = r = A_t / s = 8 / 9.0738#

#color(magenta)("Area of inscribed circle " = A_i = pi r^2 = pi * (8/9.0738)^2 ~~ 2.442#

Wyatt Anderson · Answer

undefined To find the area of the triangle's incircle, you can use the formula:

Area of incircle = semiperimeter * inradius

First, find the semiperimeter (s) of the triangle:

s = (a + b + c) / 2

where a, b, and c are the lengths of the triangle's sides. Since the lengths of the sides are not given, you need to find them using the law of sines:

a/sin(A) = b/sin(B) = c/sin(C)

Then, use the given angles to find the lengths of the sides. Once you have the lengths of the sides, you can find the semiperimeter.

Next, find the inradius (r) using the formula:

inradius = Area of triangle / semiperimeter

Once you have the inradius, multiply it by the semiperimeter to find the area of the incircle.

Noah Atkinson · Answer

undefined To find the area of the incircle of a triangle, you can use the formula:

$$A_{\text{incircle}} = \frac{s}{2} \cdot r$$

where $A_{\text{incircle}}$ is the area of the incircle, $s$ is the semi-perimeter of the triangle, and $r$ is the inradius of the triangle. The semi-perimeter $s$ can be calculated as:

$$s = \frac{a + b + c}{2}$$

where $a$, $b$, and $c$ are the lengths of the sides of the triangle. The inradius $r$ can be calculated using the formula:

$$r = \frac{A}{s}$$

where $A$ is the area of the triangle.

Given that the area of the triangle is $8$, you can use Heron's formula to find the lengths of the sides of the triangle. Then, calculate the semi-perimeter $s$. After finding $s$, use it to find the inradius $r$. Finally, use the formula for the area of the incircle to find $A_{\text{incircle}}$.