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In the circle shown below, ∠BAC is a central angle, r is the length of AB, and θ is the measure of ∠BAC in degrees.

https://lh5.googleusercontent.com/vGxgQ60PGu4WCkFdHDXFR0d0rVx_7wMLJeY3zCtQt6YKA_2JQU4UnzT12axHzV3-6MrC2TykXFsOvRbBqpUZSZYkHy_c63yjSiY3ro-DxnMDMiIauumG3bGMqMCaFV-EBNQaAKs_

Which of the following would be used to derive the formula for the area of sector BAC?

The equation that sets the ratio of θ to the length of arc BC equal to the ratio of 360 to the circumference of the circle shown above.

The equation that sets the ratio of θ to the measure of arc BC equal to the ratio of the area of sector BAC to the area of the circle shown above.

The equation that sets the ratio of the length of r to the area of the circle equal to the ratio of the area of sector BAC to the area of the circle shown above.

The equation that sets the ratio of θ to the area of sector BAC equal to the ratio of 360 to the area of the circle shown above.

 Feb 24, 2019
 #1
avatar+104628 
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The equation that sets the ratio of θ to the area of sector BAC equal to the ratio of 360 to the area of the circle shown above.

 

To see why this is so.....let the radius = 1

So the area of the circle = pi(1)^2 =  pi

 

Now....imagine that θ = 60°

 

So

 

60  / area of sector  =   360 / pi

 

60 /area of sector =  360 /pi       we can write

 

area of sector / 60 =   pi /360       multiply both sides by 60

 

area of sector =   (60 /360) pi   =    pi /6   

 

Since 60° is 1/6 of 360°.....the area should be  (1/6) pi   =  pi /6

 

 

cool cool cool

 Feb 24, 2019
edited by CPhill  Feb 24, 2019

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