1. A circle with centre O has radius r cm. A sector of the circle, which has an angle of Ɵ radians at O, has perimeter 6 cm.  show that Ɵ = 6/r - 2 , and express the area A cm^2 of the...

1. A circle with centre O has radius r cm. A sector of the circle, which has an angle of Ɵ radians at O, has perimeter 6 cm.

 show that Ɵ = 6/r - 2 , and express the area A cm^2 of the sector in terms of r. 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the equation that gives the perimeter of sector of circle of `x^o` , such that:

`p = 2r + (x^o*pi*r/180^o)`

You need to convert the radians into degrees, such that:

`x^o = (theta*180^o)/pi`

Replacing `(theta*180^o)/pi` for `x^o` yields:

`p = 2r + (theta*180^o*pi*r)/((pi*180^o))`

Reducing duplicate factors yields:

`p = 2r + theta*r`

Since the problem provides the information that perimeter `p = 6` , yields:

`6 = 2r + theta*r => r*theta = 6 - 2r => theta = (6 - 2r)/r => theta = 6/r - (2r)/r => theta = 6/r - 2`

Hence, testing if the equation `theta = 6/r - 2` holds, under the given conditions, yields that the equation `theta = 6/r - 2` is valid.

You need to evaluate the area of circle sector using the following formula, such that:

`A = (theta*r^2)/2`

Replacing `(6 - 2r)/r` for `theta` yields:

`A = ((6 - 2r)/r*r^2)/2 => A = (6r)/2 - (2r^2)/2`

`A = 3r - r^2`

Hence, evaluating the area of circle sector, in terms of r, yields `A = 3r - r^2.`

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