A triangle has vertices A, B, and C. Vertex A has an angle of #pi/6 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #4 #. What is the area of the triangle's incircle?
The area of the triangle's incircle is
By signing up, you agree to our Terms of Service and Privacy Policy
The area of the incircle of a triangle can be calculated using the formula:
[ \text{Area of incircle} = \text{semiperimeter} \times \text{inradius} ]
The semiperimeter of the triangle can be calculated as ( s = \frac{a + b + c}{2} ), where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.
Given the angles at vertices A and B, and the area of the triangle, we can find the lengths of the sides using trigonometric relations.
Let's denote the side lengths as follows:
- ( a ) is opposite angle A,
- ( b ) is opposite angle B,
- ( c ) is opposite angle C.
We can use the law of sines to find the side lengths:
[ \frac{a}{\sin(\frac{\pi}{6})} = \frac{b}{\sin(\frac{\pi}{12})} = \frac{c}{\sin(\pi - \frac{\pi}{6} - \frac{\pi}{12})} ]
Then, once the side lengths are known, we can find the semiperimeter ( s ).
Next, we need to find the inradius ( r ). The formula to calculate the inradius of a triangle is:
[ r = \frac{\text{Area}}{\text{Semiperimeter}} ]
Finally, once we have the semiperimeter and inradius, we can find the area of the incircle using the formula mentioned at the beginning.
Let's proceed with the calculations:
Given: [ \text{Angle A} = \frac{\pi}{6} ] [ \text{Angle B} = \frac{\pi}{12} ] [ \text{Area of triangle} = 4 ]
We can use trigonometric relations to find the side lengths: [ a = 2 \times \sin(\frac{\pi}{6}) ] [ b = 2 \times \sin(\frac{\pi}{12}) ]
Then, calculate the semiperimeter: [ s = \frac{a + b + c}{2} ]
Find the inradius: [ r = \frac{\text{Area}}{s} ]
Finally, calculate the area of the incircle: [ \text{Area of incircle} = s \times r ]
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.
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.
- The area of a regular hexagon is #38# #cm^2#. What is the area of a regular hexagon with sides four times as long?
- A circle has a chord that goes from #( 5 pi)/6 # to #(5 pi) / 4 # radians on the circle. If the area of the circle is #18 pi #, what is the length of the chord?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/8 #, and the triangle's area is #14 #. What is the area of the triangle's incircle?
- A triangle has sides with lengths of 2, 7, and 6. What is the radius of the triangles inscribed circle?
- A triangle has corners at #(9 ,3 )#, #(4 ,6 )#, and #(2 ,4 )#. What is the area of the triangle's circumscribed circle?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7