22. Circles
A circle is a set of points equidistant from a point called the center.
The distance of any point of the circle from the center is called the radius.
Twice the radius is a measure referred to as a diameter.
A cicle contains 360°. Area = $πr^2$, Perimeter (also called circumference)=$2πr$
Any smaller "slice" of the circle bounded by part of the circumference on one edge, and the two segments connecting its end points to the center on the others, is called a sector of the circle.
The area of the sector is the central angle of the sector divided by 360° times the total area of the circle.
The perimeter of the sector is the central angle of the sector divided by 360° times the total perimeter of the circle, + 2r.

E.g. Charlie slices a pie into 4 pieces each with a central angle double the last. What is the area of the smallest piece of pie?
Let the central sector angles be x, 2x, 4x, and 8x respectively. So $15x=360°$ this gives $x=24°$
So area of the smallest slice of pie=$πr^2*24/360={πr^2}/15$

Any part of the circumference of the circle is called an arc.
If the arc is smaller than half the total circumference, it "contains" <180° and is called a minor arc.
An arc that is larger than half the total circumference is called a major arc.

The angle formed between the end-points of the arc and a point is said to be the angle subtended by the arc at that point.
We are mostly interested in angles subtended by an arc at the center of the circle, and to any point on the circumference.

From the figure above, O is the center of the circle. Minor arc BC subtends angle BOC at the center.
The same minor arc subtends an angle BAC at point A on the circumference.
Angle BOC = 2*Angle BAC


In the above figure, AOB is a diameter of the circle. A and B are diametrically opposite points on the circumference.
C and D are any arbitrary points on arc ACDB, which is a semi-circle.
Now, angle ACB, and angle ADB are both right angles, i.e. each measures 90°.

 


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