A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(3pi)/4 #, and the triangle's area is #8 #. What is the area of the triangle's incircle?

Answer 1

The area of the incircle is #=2.14u^2#

The area of the triangle is #A=8#

The angle #hatA=1/12pi#

The angle #hatB=3/4pi#

The angle #hatC=pi-(1/12pi+3/4pi)=1/6pi#

The sine rule is

#a/(sin hat (A))=b/sin hat (B)=c/sin hat (C)=k#

So,

#a=ksin hatA#

#b=ksin hatB#

#c=ksin hatC#

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=csin hatB#

So,

#A=1/2ksin hatA*csin hatB=1/2ksin hatA*ksin hatC*sin hatB#

#A=1/2k^2*sin hatA*sin hatB*sin hatC#

#k^2=(2A)/(sin hatA*sin hatB*sin hatC)#

#k=sqrt((2A)/(sin hatA*sin hatB*sin hatC))#

#=sqrt(16/(sin(1/12pi)*sin(3/4pi)*sin(1/6pi)))#

#=13.22#

Therefore,

#a=13.22sin(1/12pi)=3.42#

#b=13.22sin(3/4pi)=9.35#

#c=13.22sin(1/6pi)=6.61#

The radius of the incircle is #=r#

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

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

#=16/(19.38)=0.83#

The area of the incircle is

#area=pi*r^2=pi*0.83^2=2.14u^2#

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Answer 2

To find the area of the triangle's incircle, we first need to calculate the semi-perimeter (( s )) of the triangle using the formula:

[ s = \frac{a + b + c}{2} ]

where ( a ), ( b ), and ( c ) are the lengths of the triangle's sides.

Then, we can use Heron's formula to find the area (( A )) of the triangle:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

Next, we can use the formula for the radius (( r )) of the incircle of a triangle:

[ r = \frac{A}{s} ]

where ( A ) is the area of the triangle and ( s ) is the semi-perimeter.

Finally, we can find the area of the triangle's incircle using the formula for the area of a circle:

[ \text{Area of incircle} = \pi \cdot r^2 ]

Substitute the values obtained into the formula to find the area of the incircle.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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