# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/3 #, vertex B has an angle of #(5 pi)/12 #, and the triangle's area is #16 #. What is the area of the triangle's incircle?

Area of I’m circle

Given

Area of triangle

Similarly,

Semi perimeter

In radius

Area of in circle

By signing up, you agree to our Terms of Service and Privacy Policy

To find the area of the triangle's incircle, we need to use the formula ( A = rs ), where ( A ) is the area of the triangle, ( r ) is the radius of the incircle, and ( s ) is the semi-perimeter of the triangle. The semi-perimeter is calculated as ( s = \frac{a + b + c}{2} ), where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.

The lengths of the sides of the triangle can be found using the Law of Sines, which states that ( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ). In this case, we know two angles and the area of the triangle, so we can find the third angle using the fact that the sum of the angles in a triangle is ( \pi ).

Once we have the lengths of the sides, we can calculate the semi-perimeter and then use it along with the area of the triangle to find the radius of the incircle. Finally, we use the formula for the area of a circle (( A = \pi r^2 )) to find the area of the incircle.

Let's proceed with these calculations:

Given:

- Angle ( A = \frac{\pi}{3} )
- Angle ( B = \frac{5\pi}{12} )
- Area of the triangle ( A = 16 )

Using the fact that the sum of the angles in a triangle is ( \pi ), we can find the third angle: [ C = \pi - \left( \frac{\pi}{3} + \frac{5\pi}{12} \right) ] [ C = \pi - \frac{4\pi}{12} - \frac{5\pi}{12} ] [ C = \pi - \frac{9\pi}{12} ] [ C = \frac{3\pi}{12} ] [ C = \frac{\pi}{4} ]

Now, we can use the Law of Sines to find the lengths of the sides of the triangle. Let ( a ), ( b ), and ( c ) be the lengths of the sides opposite angles ( A ), ( B ), and ( C ), respectively: [ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

We can choose any convenient scale for these lengths, so let's assume: [ a = 4x, \quad b = 4y, \quad c = 4z ]

[ \frac{4x}{\sin\left(\frac{\pi}{3}\right)} = \frac{4y}{\sin\left(\frac{5\pi}{12}\right)} = \frac{4z}{\sin\left(\frac{\pi}{4}\right)} ]

Now, we can solve for ( x ), ( y ), and ( z ).

[ x = \frac{4 \sin\left(\frac{\pi}{3}\right)}{\sin\left(\frac{\pi}{3}\right)} = 4 ]

[ y = \frac{4 \sin\left(\frac{5\pi}{12}\right)}{\sin\left(\frac{\pi}{3}\right)} ]

[ z = \frac{4 \sin\left(\frac{\pi}{4}\right)}{\sin\left(\frac{\pi}{3}\right)} ]

We can calculate ( y ) and ( z ):

[ y \approx 4.784 ] [ z \approx 5.657 ]

The semi-perimeter ( s ) is: [ s = \frac{a + b + c}{2} = \frac{4x + 4y + 4z}{2} ]

[ s = \frac{4(4) + 4(4.784) + 4(5.657)}{2} ]

[ s \approx \frac{64.456}{2} ]

[ s \approx 32.228 ]

Now, using the formula ( A = rs ), we can solve for ( r ): [ 16 = r \times 32.228 ]

[ r = \frac{16}{32.228} ]

[ r \approx 0.496 ]

Finally, we use the formula for the area of a circle to find the area of the incircle: [ A_{\text{incircle}} = \pi r^2 ]

[ A_{\text{incircle}} = \pi \times (0.496)^2 ]

[ A_{\text{incircle}} \approx 0.386 ]

Therefore, the area of the triangle's incircle is approximately ( 0.386 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

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.

- Find the equation to the circle which touches y-axis at the origin and passes though the point (alpha,beta)?
- A triangle has corners at #(2 , 6 )#, #(4 ,8 )#, and #(1 ,5 )#. What is the radius of the triangle's inscribed circle?
- A triangle has corners at #(2 , 6 )#, #(3 ,9 )#, and #(4 ,5 )#. What is the radius of the triangle's inscribed circle?
- A circle has a chord that goes from #( 3 pi)/2 # to #(7 pi) / 4 # radians on the circle. If the area of the circle is #121 pi #, what is the length of the chord?
- Two circles have the following equations #(x +2 )^2+(y -5 )^2= 16 # and #(x +4 )^2+(y +1 )^2= 49 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7