A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/6 #, and the triangle's area is #4 #. 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 #pi/6  #, and the triangle's area is #4 #.  What is the area of the triangle's incircle?

Claire Akin · Accepted Answer

#1.073\ ext{unit}^2# Given that in #\Delta ABC#, #A=\pi/12#, #B=\pi/6# #C=\pi-A-B# #=\pi-\pi/12-\pi/6# #={3\pi}/4# 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 (\pi/6)}=\frac{c}{\sin ({3\pi}/4)}=k\ ext{let}# #a=k\sin(\pi/12)=0.259k# # b=k\sin(\pi/6)=0.5k# #c=k\sin({3\pi}/4)=0.707k# #s=\frac{a+b+c}{2}# #=\frac{0.259k+0.5k+0.707k}{2}=0.733k# Area of #\Delta ABC# from Hero's formula #\Delta=\sqrt{s(s-a)(s-b)(s-c)}# #4=\sqrt{0.733k(0.733k-0.259k)(0.733k-0.5k)(0.733k-0.707k)}# #4=0.0459k^2# #k^2=87.187# Now, the in-radius (#r#) of #\Delta ABC# #r=\frac{\Delta}{s}# #r=\frac{4}{0.733k}# Hence, the area of inscribed circle of #\Delta ABC# #=\pi r^2# #=\pi (4/{0.733k})^2# #=\frac{16\pi}{0.537289k^2}# #=\frac{93.5539}{87.187}\quad (\because k^2=87.187)# #=1.073\ ext{unit}^2#

Jayden Jackson · Answer

undefined To find the area of the triangle's incircle, we first need to determine the triangle's side lengths using the given angles and area.

We use the formula for the area of a triangle:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Knowing that the area of the triangle is 4, we can set up the equation:

$$ 4 = \frac{1}{2} \times \text{base} \times \text{height} $$

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

After finding the side lengths, we can calculate the semi-perimeter (s) of the triangle using the formula:

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

Then, we can find the radius of the incircle (r) using Heron's formula:

$$ \text{Area} = \sqrt{s \times (s - a) \times (s - b) \times (s - c)} $$

Once we have the radius of the incircle, we can find its area using the formula:

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