A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/8 #, and the triangle's area is #16 #. 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/2 #, vertex B has an angle of #( pi)/8  #, and the triangle's area is #16 #.  What is the area of the triangle's incircle?

Wyatt Anderson · Accepted Answer

Area of in circle # A_i = 6.7# sq units

#A_t = (1/2) ab sin C =(1/2)b c sin A = (1/2) c a sin B# #hat A = pi/2, hat B = pi/8, hat C = (3pi)/8# #a b = (2 * 16) / sin ((3pi)/8) = 34.64# #c a = (2*16)/ sin (pi/8) = 83.62# #b c= (2*16)/ sin(pi/2) = 32# #a = (abc) / (b c) = sqrt(34.64*83.62*32) / 32 = 9.51# #b = (abc) / (c a) = sqrt(34.64* 83.62*32) / 83.62 = 3.64# #c = sqrt(34.64*83.62*32)/34.64 = 8.79# Semi perimeter #s = (a + b + c) / 2# #s = (9.51 + 3.64 + 8.79) / 2 = 10.97# Radius of in circle # r = A_t / s = 16 / 10.97 = 1.46# Area of in circle # A_i = pi r^2 = pi * 1.46^2 = 6.7#

Sadie Allen · Answer

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

$$ \text{Area of incircle} = \pi \times r^2 $$

where $ r $ is the radius of the incircle. The radius of the incircle can be found using the formula:

$$ r = \frac{\text{Area of triangle}}{\text{Semiperimeter of triangle}} $$

The semiperimeter of the triangle, denoted as $ s $, is half the sum of the lengths of its sides. Since we have the angles of the triangle, we can use trigonometric relationships to find the side lengths.

Given that angle $ A = \frac{\pi}{2} $ and angle $ B = \frac{\pi}{8} $, we can find angle $ C $ as $ \pi - \frac{\pi}{2} - \frac{\pi}{8} = \frac{5\pi}{8} $. This makes it a right triangle.

Now, using trigonometric ratios:

$$ \frac{BC}{AB} = \tan(\frac{\pi}{8}) $$

Given that the area of the triangle is 16, we can now solve for the side lengths $ AB $ and $ BC $. Once we have the side lengths, we can find the semiperimeter and then the radius of the incircle, and finally the area of the incircle.

If you need further clarification or assistance with any step, feel free to ask.