# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/8 #, and the triangle's area is #16 #. What is the area of the triangle's incircle?

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, you can use the formula:

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

where ( r ) is the radius of the incircle. The radius of the incircle can be found using the formula:

[ r = \frac{\text{Area of triangle}}{\text{Semiperimeter of triangle}} ]

The semiperimeter of the triangle, denoted as ( s ), is half the sum of the lengths of its sides. Since we have the angles of the triangle, we can use trigonometric relationships to find the side lengths.

Given that angle ( A = \frac{\pi}{2} ) and angle ( B = \frac{\pi}{8} ), we can find angle ( C ) as ( \pi - \frac{\pi}{2} - \frac{\pi}{8} = \frac{5\pi}{8} ). This makes it a right triangle.

Now, using trigonometric ratios:

[ \frac{BC}{AB} = \tan(\frac{\pi}{8}) ]

Given that the area of the triangle is 16, we can now solve for the side lengths ( AB ) and ( BC ). Once we have the side lengths, we can find the semiperimeter and then the radius of the incircle, and finally the area of the incircle.

If you need further clarification or assistance with any step, feel free to ask.

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.

- The diameter of a U.S quarter is 24.26 millimeters. How long is a line of 37 quarters laid edge to edge?
- What is the equation of the circle with a center at #(3 ,1 )# and a radius of #1 #?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(3pi)/8 #, and the triangle's area is #6 #. What is the area of the triangle's incircle?
- A circle has a chord that goes from #( 5 pi)/3 # to #(17 pi) / 12 # radians on the circle. If the area of the circle is #18 pi #, what is the length of the chord?
- A triangle has sides with lengths of 3, 9, and 8. What is the radius of the triangles inscribed circle?

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