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A circle is the set of all points on a plane that have the same distance to a fixed point. The fixed point is called the center of the circle, while the same distance is called the radius.
Mathematically, a circle with center and radius can be expressed with the equation:
How is the visualization of the circle equation?
Important elements of circle:
Arc of circle is a part of the circumference of the circle that is bounded by two points on the circle. Arc is denoted with a curved symbol above the letters, for example .
Types of arc based on their length:
Central angle is an angle whose vertex is at the center of the circle and whose sides are radii of the circle.
Properties of central angle:
Inscribed angle is an angle whose vertex is on the circle and whose sides are chords.
If central angle and inscribed angle subtend the same arc, then the measure of inscribed angle is half the measure of central angle.
Example application:
Arc length is directly proportional to the measure of central angle that subtends it.
Where:
Sector is the region bounded by two radii and an arc of circle.
We can visualize the sector area using the equation above.
A circle has a radius of . If the central angle that subtends an arc is , determine:
Solution:
Given: ,
Arc length:
Sector area:
A circle has a diameter of . If an inscribed angle that subtends an arc is , determine the measure of central angle that subtends the same arc!
In a circle with center and radius , there is an arc with central angle . Calculate:
Two inscribed angles subtend the same arc. If one inscribed angle measures , determine the measure of the other inscribed angle!
Given: ,
