A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/12 #, 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/12 #, vertex B has an angle of #(5pi)/12  #, and the triangle's area is #12 #.  What is the area of the triangle's incircle?

Mason Adams · Accepted Answer

#3.8075\ ext{unit}^2# Given that in #\Delta ABC#, #A=\pi/12#, #B={5\pi}/12# #C=\pi-A-B# #=\pi-\pi/12-{5\pi}/12# #={\pi}/2# 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 ({5\pi}/12)}=\frac{c}{\sin ({\pi}/2)}=k\ ext{let}# #a=k\sin(\pi/12)=0.259k# # b=k\sin({5\pi}/12)=0.966k# #c=k\sin({\pi}/2)=k# #s=\frac{a+b+c}{2}# #=\frac{0.259k+0.966k+k}{2}=1.1125k# Area of #\Delta ABC# from Hero's formula #\Delta=\sqrt{s(s-a)(s-b)(s-c)}# #12=\sqrt{1.1125k(1.1125k-0.259k)(1.1125k-0.966k)(1.1125k-k)}# #12=0.125k^2# #k^2=96# Now, the in-radius (#r#) of #\Delta ABC# #r=\frac{\Delta}{s}# #r=\frac{12}{1.1125k}# Hence, the area of inscribed circle of #\Delta ABC# #=\pi r^2# #=\pi (12/{1.1125k})^2# #=\frac{144\pi}{1.2376k^2}# #=\frac{365.521}{96}\quad (\because k^2=96)# #=3.8075\ ext{unit}^2#

Aurora Ayala · Answer

undefined The area of the triangle's incircle is $6\sqrt{3} - 3\pi$.

Caroline Aucoin · Answer

undefined The area of a triangle can be expressed in terms of its semiperimeter (s) and the radii (r) of its incircle (r) and circumcircle (R) using the formula:

Area = s * r

where s = (a + b + c) / 2 and a, b, c are the lengths of the sides of the triangle.

The semiperimeter (s) can be found by adding the lengths of the sides of the triangle and dividing by 2:

s = (AB + BC + CA) / 2

To find the lengths of the sides AB, BC, and CA, you can use the law of sines:

AB = 2R * sin(A)
BC = 2R * sin(B)
CA = 2R * sin(C)

Given that the area of the triangle is 12 and the angles at vertices A and B are π/12 and (5π)/12 respectively, you can use these angles to calculate the lengths of the sides AB, BC, and CA. Once you have the lengths of the sides, you can find the semiperimeter (s) and then use it along with the area to find the radius of the incircle (r).

Finally, once you have the radius of the incircle (r), you can calculate the area of the incircle using the formula for the area of a circle:

Area of incircle = π * r^2