A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{8}$ $pi/8$ , vertex B has an angle of $\frac{π}{12}$ $(pi)/12$ , and the triangle's area is $1$ $1$ . What is the area of the triangle's incircle?

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A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{8}$ $pi/8$ , vertex B has an angle of $\frac{π}{12}$ $(pi)/12$ , and the triangle's area is $1$ $1$ . What is the area of the triangle's incircle?

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Answer 1 · 2018-02-01T14:53:26

Area of triangle’s Incircle $A_{i} = 0.2424$ $A_i = color(green)0.2424$

Explanation:

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Three angles are $A = \frac{π}{8}, B = \frac{π}{12}, C = π - \frac{π}{8} - \frac{π}{12} = \frac{19 π}{24}$ $A = pi / 8, B = pi / 12, C = pi - pi/8 - pi/12 = (19pi)/24$

Given Area of triangle A_t = 1#

$b c = \frac{A_{t}}{(\frac{1}{2}) sin A} = \frac{2 \cdot 1}{sin (\frac{π}{8})} = 5.2263$ $bc = A_t / ((1/2) sin A) = (2 * 1) / sin (pi / 8) = 5.2263$

$c a = \frac{A_{t}}{(\frac{1}{2}) sin B} = \frac{2}{sin (\frac{π}{12})} = 7.7274$ $ca = A_t / ((1/2) sin B) = 2 / sin (pi / 12) = 7.7274$

$a b = \frac{A_{t}}{(\frac{1}{2}) sin C} = \frac{2}{sin (\frac{19 π}{24})} = 3.2854$ $ab = A_t / ((1/2) sin C) = 2 / sin ((19pi)/24) = 3.2854$

$\sqrt{a b \cdot b c \cdot c a} = (a b c) = \sqrt{5.2263 \cdot 7.7274 \cdot 3.2854} = 11.5188$ $sqrt(ab * bc * ca)= (abc) = sqrt(5.2263 * 7.7274 * 3.2854) = 11.5188$

$a = \frac{a b c}{b c} = \frac{11.5188}{5.2263} = 2.204$ $a = (abc) / (bc) = 11.5188 / 5.2263 = 2.204$

$b = \frac{a b c}{c a} = \frac{11.5188}{7.7274} = 1.4906$ $b = (abc) / (ca) = 11.5188 / 7.7274 = 1.4906$

$c = \frac{a b c}{A B} = \frac{11.5188}{3.2854} = 3.5061$ $c = (abc) / (AB) = 11.5188 / 3.2854 = 3.5061$

Semi perimeter $s = \frac{a + b + c}{2} = \frac{2.204 + 1.4906 + 3.5061}{2} \approx 3.6$ $s = (a + b + c) / 2 = (2.204 + 1.4906 + 3.5061) / 2 ~~ 3.6$

Inradius $I_{r} = \frac{A_{t}}{\frac{s}{2}} = \frac{1}{3.6}$ $I_r = A_t / (s/2) = 1 / 3.6$

Area of Incircle $A_{i} = π {(I_{r})}_{r}^{2} = π \cdot {(\frac{1}{3.6})}^{2} = 0.2424$ $A_i = pi (I_r)^2 = pi * (1/3.6)^2 = color(green)(0.2424)$

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A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{8}$ $pi/8$ , vertex B has an angle of $\frac{π}{12}$ $(pi)/12$ , and the triangle's area is $1$ $1$ . What is the area of the triangle's incircle?

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A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{8}$ $pi/8$ , vertex B has an angle of $\frac{π}{12}$ $(pi)/12$ , and the triangle's area is $1$ $1$ . What is the area of the triangle's incircle?