# 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 #99 pi #, what is the length of the chord?

7.62 units

First, use a unit circle to determine the end points of the chord on the circle.

The angle between the two equivalent sides of the triangle is equal to the difference between the angles given in the problem:

Finally, the law of cosines can be used to determine an equation for the length of the chord:

Since both

The problem states that the area of the circle is

Plug this value into the equation for the chord:

Note: Since the units of length are not provided, just use "units."

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

To find the length of the chord in a circle given its central angle and the area of the circle, you can use the formula:

[ \text{Length of chord} = 2 \times \text{radius} \times \sin\left(\frac{\theta}{2}\right) ]

Where:

- ( \theta ) is the central angle in radians,
- ( \text{radius} ) is the radius of the circle.

First, we need to find the radius of the circle using the given area:

[ \text{Area of circle} = \pi r^2 ] [ 99\pi = \pi r^2 ] [ r^2 = 99 ] [ r = \sqrt{99} ]

Now, we can calculate the length of the chord using the given central angle:

[ \theta = \left(\frac{7\pi}{4}\right) - \left(\frac{3\pi}{2}\right) = \frac{\pi}{4} ]

[ \text{Length of chord} = 2 \times \sqrt{99} \times \sin\left(\frac{\pi}{8}\right) ]

Using a calculator, we can find the sine value of ( \frac{\pi}{8} ), which is approximately 0.3827.

[ \text{Length of chord} = 2 \times \sqrt{99} \times 0.3827 ]

[ \text{Length of chord} \approx 2 \times \sqrt{99} \times 0.3827 ]

[ \text{Length of chord} \approx 2 \times 9.9498 \times 0.3827 ]

[ \text{Length of chord} \approx 7.575 ]

So, the length of the chord is approximately 7.575 units.

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.

- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/4 #, and the triangle's area is #36 #. What is the area of the triangle's incircle?
- A triangle has corners at #(2 ,5 )#, #(3 ,1 )#, and #(4 ,2 )#. What is the area of the triangle's circumscribed circle?
- Points #(5 ,4 )# and #(2 ,0 )# are #(5 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?
- Find the points of intersection of line #y=3+2x# with the circle #x^2+y^2=50#?
- A circle has a center that falls on the line #y = 2/3x +7 # and passes through # ( 3 ,4 )# and #(6 ,4 )#. What is the equation of the circle?

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