A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{12}$ $pi/12$ , vertex B has an angle of $\frac{π}{8}$ $pi/8$ , and the triangle's area is $4$ $4$ . 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 $\frac{π}{12}$ $pi/12$ , vertex B has an angle of $\frac{π}{8}$ $pi/8$ , and the triangle's area is $4$ $4$ . What is the area of the triangle's incircle?

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Answer 1 · 2018-06-10T08:04:37

$Area of incircle A_{r} = π r^{2} = 0.00166 sq units$ $color(purple)("Area of incircle " A_r = pi r^2 = 0.00166 " sq units"$

Explanation:

The distances from the incenter to each side are equal to the inscribed circle's radius. The area of the triangle is equal to 12×r×(the triangle's perimeter), 1 2 × r × ( the triangle's perimeter ) , where r is the inscribed circle's radius.

First to find the perimeter of the triangle.

$ˆ A = \frac{π}{12}, ˆ B = \frac{π}{8}, ˆ C = \frac{19 π}{24}, A_{t} = 4$ $hat A = pi/12, hat B = pi/8, hat C = (19pi)/24, A_t = 4$

Area of triangle $A_{t} = (\frac{1}{2}) a b sin C = (\frac{1}{2}) b c sin A = (\frac{1}{2}) c a sin B$ $A_t = (1/2)ab sin C = (1/2)bc sin A = (1/2)ca sin B$

$a b = \frac{4 \cdot 2}{sin (\frac{19}{24})} = 13.14$ $ab = (4 * 2) / sin ((19)/24) = 13.14$

Similarly, $b c = \frac{4 \cdot 2}{sin (\frac{π}{12})} = 30.91$ $bc = (4 * 2) / sin (pi/12) = 30.91$

$c a = \frac{4 \cdot 2}{sin (\frac{π}{8})} = 20.91$ $ca = (4 * 2) / sin (pi/8) = 20.91$

$a b \cdot b c \cdot c a = {(a b c)}^{2} = 13.14 \cdot 30.91 \cdot 20.91$ $ab * bc * ca = (abc)^2 = 13.14 * 30.91 * 20.91$

$a b c = \sqrt{13.14 \cdot 30.91 \cdot 20.91}$ $abc = sqrt(13.14 * 30.91 * 20.91)$

$a = \frac{a b c}{b c} = \frac{\sqrt{13.14 \cdot 30.91 \cdot 20.91}}{30.91} = 2.98$ $a = (abc) / (bc) = sqrt(13.14 * 30.91 * 20.91) / 30.91 = 2.98$

Likewise, $b = \frac{a b c}{c a} = \frac{\sqrt{13.14 \cdot 30.91 \cdot 20.91}}{20.91} = 4.41$ $b = (abc) / (ca) = sqrt(13.14 * 30.91 * 20.91) / 20.91 = 4.41$

$c = \frac{a b c}{a b} = \frac{\sqrt{13.14 \cdot 30.91 \cdot 20.91}}{13.14} = 7.01$ $c = (abc) / (ab) = sqrt(13.14 * 30.91 * 20.91) / 13.14 = 7.01$

$Perimeter of the triangle p = a + b + c = 2.98 + 4.41 + 7.01 = 14.4$ $"Perimeter of the triangle " p = a + b + c = 2.98 + 4.41 + 7.01 = 14.4$

$Radius of incircle r = \frac{A_{t}}{12 \cdot p} = \frac{4}{12 \cdot 14.4} = 0.023$ $"Radius of incircle " r = A_t / (12 * p) = 4 / (12 * 14.4) = 0.023$

$Area of incircle A_{r} = π r^{2} = π \cdot {0.023}^{2} = 0.00166$ $color(purple)("Area of incircle " A_r = pi r^2 = pi * 0.023^2 = 0.00166$

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A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{12}$ $pi/12$ , vertex B has an angle of $\frac{π}{8}$ $pi/8$ , and the triangle's area is $4$ $4$ . What is the area of the triangle's incircle?

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Explanation:

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A triangle has vertices A, B, and C. Vertex A has an angle of $\frac{π}{12}$ $pi/12$ , vertex B has an angle of $\frac{π}{8}$ $pi/8$ , and the triangle's area is $4$ $4$ . What is the area of the triangle's incircle?