A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(7pi)/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 #(7pi)/8  #, and the triangle's area is #8 #.  What is the area of the triangle's incircle?

Violet Aldrich · Accepted Answer

#1.0767\ ext{unit}^2# Given that in #\Delta ABC#, #A=\pi/12#, #B={7\pi}/8# #C=\pi-A-B# #=\pi-\pi/12-{7\pi}/8# #={\pi}/24# from sine in #\Delta ABC#, we have #\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}# #\frac{a}{\sin(\pi/12)}=\frac{b}{\sin ({7\pi}/8)}=\frac{c}{\sin ({\pi}/24)}=k\ ext{let}# #a=k\sin(\pi/12)=0.259k# # b=k\sin({7\pi}/8)=0.383k# #c=k\sin({\pi}/24)=0.1305k# #s=\frac{a+b+c}{2}# #=\frac{0.259k+0.383k+0.1305k}{2}=0.38625k# Area of #\Delta ABC# from Hero's formula #\Delta=\sqrt{s(s-a)(s-b)(s-c)}# #8=\sqrt{0.38625k(0.38625k-0.259k)(0.38625k-0.383k)(0.38625k-0.1305k)}# #8=0.006392k^2# #k^2=1251.634# Now, the in-radius (#r#) of #\Delta ABC# #r=\frac{\Delta}{s}# #r=\frac{8}{0.38625k}# Hence, the area of inscribed circle of #\Delta ABC# #=\pi r^2# #=\pi (8/{0.38625k})^2# #=\frac{64\pi}{0.1492k^2}# #=\frac{1347.699}{1251.634}\quad (\because k^2=1251.634)# #=1.0767\ ext{unit}^2#

Sadie Allen · Answer

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

$$ A = rs $$

Where:
- A is the area of the incircle,
- r is the radius of the incircle,
- s is the semi-perimeter of the triangle.

To find the semi-perimeter (s) of the triangle, you add the lengths of its three sides and divide by 2:

$$ s = \frac{{\text{Side 1} + \text{Side 2} + \text{Side 3}}}{2} $$

The lengths of the sides can be found using the law of cosines:

$$ \text{Side} = \sqrt{a^2 + b^2 - 2ab\cos(\theta)} $$

Where:
- a and b are the lengths of two sides of the triangle,
- θ is the angle between those two sides.

Once you have the lengths of the sides and the semi-perimeter, you can use Heron's formula to find the area of the triangle:

$$ A = \sqrt{s(s - \text{Side 1})(s - \text{Side 2})(s - \text{Side 3})} $$

After finding the area of the triangle, you can calculate the radius of the incircle using the formula mentioned earlier, and then multiply it by the semi-perimeter to find the area of the incircle.