A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/8 #, and the triangle's area is #15 #. 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)/8  #, and the triangle's area is #15 #.  What is the area of the triangle's incircle?

Jayden Jackson · Accepted Answer

The area of the incircle is #=2.29u^2#

The area of the triangle is #A=15#

The angle #hatA=1/12pi#

The angle #hatB=5/8pi#

The angle #hatC=pi-(5/8pi+1/12pi)=7/24pi#

The sine rule is

#a/sinA=b/sinB=c/sinC=k#

So,

#a=ksinA#

#b=ksinB#

#c=ksinC#

Let the height of the triangle be #=h# from the vertex #A# to the opposite side #BC#

The area of the triangle is

#A=1/2a*h#

But,

#h=csinB#

So,

#A=1/2ksinA*csinB=1/2ksinA*ksinC*sinB#

#A=1/2k^2*sinA*sinB+sinC#

#k^2=(2A)/(sinA*sinB*sinC)#

#k=sqrt((2A)/(sinA*sinB*sinC))#

#=sqrt(15/(sin(pi/12)*sin(5/8pi)*sin(7/24pi)))#

#=3.87/0.436=8.89#

Therefore,

#a=8.89sin(1/12pi)=2.3#

#b=8.89sin(5/8pi)=8.21#

#c=8.89sin(7/24pi)=7.05#

The radius of the in circle is #=r#

#1/2*r*(a+b+c)=A#

#r=(2A)/(a+b+c)#

#=15/(17.6)=0.85#

The area of the incircle is

#area=pi*r^2=pi*0.85^2=2.29u^2#

Mason Adams · Answer

undefined To find the area of the triangle's incircle, we first need to find the lengths of the sides of the triangle. We can use the Law of Sines to find the lengths of the sides.

The Law of Sines states:

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

Given the angles A and B, we have:

$ A = \frac{\pi}{12} $ and $ B = \frac{5\pi}{8} $

Since the sum of angles in a triangle is $ \pi $, we can find angle C:

$ C = \pi - A - B $

Once we have the three angles, we can use the Law of Sines to find the ratio of the sides and then find the actual lengths of the sides using the given area.

After finding the lengths of the sides, we can calculate the semi-perimeter of the triangle, $ s $, and then use Heron's formula to find the area of the triangle.

Finally, with the area of the triangle known, we can use the formula for the area of a circle ($ \pi r^2 $) to find the area of the incircle, where $ r $ is the radius of the incircle. The radius $ r $ can be found using the formula for the area of a triangle in terms of its inradius $ r $, which is $ A = rs $.