A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #( pi)/3 #, 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)/3  #, and the triangle's area is #12 #.  What is the area of the triangle's incircle?

Aaliyah Anderson · Accepted Answer

#color(brown)("Area of inscribed circle " A_r = 0.0086 " sq units"# The distances from the incenter to each side are equal to the inscribed circle's radius. The area of the triangle is equal to #color(crimson)("12×r×(the triangle's perimeter), 1 2 × r × ( the triangle's perimeter ) "#, where r is the inscribed circle's radius. #"Area of Triangle " A_t = (ab sin C) / 2 = (bc sin A) / 2 = (ca sin B) / 2# #hat A = pi/8, hat b = pi/3, hat C = (13pi)/24, A_t = 12# #ab = (2 * A_t) / sin C = (2 * 12) / sin ((13pi)/24) = 24.21# #bc = (2 * A_t) / sin A = (2 * 12) / sin (pi/8) = 62.72# #ca = (2 * A_t) / sin B = (2 * 12) / sin (pi/3) = 27.71# #a = sqrt(ab * bc * ca) / (bc) = sqrt(24.21 * 62.72 * 27.71) / 62.72 = 3.27# #b = sqrt(ab * bc * ca) / (bc) = sqrt(24.21 * 62.72 * 27.71) / 27.71 = 7.4# #c = sqrt(ab * bc * ca) / (ab) = sqrt(24.21 * 62.72 * 27.71) / 24.21 = 8.47# #"Perimeter " p = a + b + c = 3.27 + 7.4 + 8.47 = 19.14# #Incircle radius " r = A_t / (12 * p) = 12 / (12 * 19.14) = 0.0522# #color(brown)("Area of inscribed circle " A_r = pi r^2 = pi * 0.0522^2 = 0.0086 " sq units"#

Avery Allen · Answer

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

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

where:
- $ s $ is the semi-perimeter of the triangle (the sum of the lengths of its sides divided by 2)
- $ r $ is the radius of the incircle

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

Using the Law of Sines, we have:

$$ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} $$

Given:
- Angle $ A = \frac{\pi}{8} $
- Angle $ B = \frac{\pi}{3} $

We can use these angles to find the lengths of the sides, considering any suitable unit for measurement.

Let's assume $ a $ is opposite to angle $ A $, $ b $ is opposite to angle $ B $, and $ c $ is opposite to angle $ C $.

Thus, we can find the lengths of sides $ a $ and $ b $:

$$ a = \frac{b \cdot \sin(A)}{\sin(B)} $$
$$ b = \frac{a \cdot \sin(B)}{\sin(A)} $$

Now, we can find the length of side $ c $ using the fact that the sum of angles in a triangle is $ \pi $:

$$ C = \pi - A - B $$

$$ c = \frac{a}{\sin(A)} $$

Now that we have the lengths of the sides, we can find the semi-perimeter:

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

Then, we can calculate the radius of the incircle using Heron's formula for the area of the triangle:

$$ A = \sqrt{s(s-a)(s-b)(s-c)} $$

Once we have the radius, we can find the area of the incircle using the formula mentioned initially.