How do you derive the equation of a circle? How do you identify the center and radius of a circle? How do you define the radian measure of an angle? How are arc length and area of a sector related to proportionality?Please use complete sentences

Question: How do you derive the equation of a circle?
To derive the equation of a circle we just need to make use of the distance formula. In a circle, we have a center and the x and y coordinates of the circle itself. The distance between the points of the circle and the center is referred to as the radius. Let's use the coordinates (h,k) to represent the center. Applying the distance formula we'll get:

[tex]r= \sqrt{ (x-h)^{2} + (y-k)^{2} } [/tex]
[tex]r^{2}= (x-h)^{2} + (y-k)^{2} [/tex]

This is the equation of the circle.

Question: How do you identify the center and radius of a circle?
This question sounds like the inverse of the previous one. To answer this we just need to know how to interpret the equation of the circle. In the previous item, we identified (h,k) as the coordinates of the center therefore h and k would still determine the x and y coordinates of the center respectively. For the radius, the square root of the constant value to the left (denoted as r in the previous item) would be its value.

Question: How do you define the radian measure of an angle? 
The radian measure of an angle is defined as the distance traveled by the central angle divided by the circle's radius. To know the value of the radian measure, we can perform two methods. First is to measure the circle's radius and the arc length traversed by the central angle. The other one is to get the central angle in degrees and convert it to radians by using the conversion factor π/180 since 180 degrees is equal to π radians.

Question: How are arc length and area of a sector related to proportionality?
The arc length and area of a sector are related to proportionality because their formulas are derived by using proportions. For both, we just basically get the ratio of the selected length or area over the total length (circumference) or area of the circle (2πr).

For arc length:
[tex] \frac{length}{2 \pi r} = \frac{central.angle}{360.degrees} [/tex]

For area of a sector:
[tex] \frac{area}{\pi r^{2}} = \frac{central.angle}{360.degrees} [/tex]