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10-6 Circles and Arcs

10-6 Circles and Arcs. Circles. In a plane, a circle is the set of all points equidistant from a given point called the center .

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10-6 Circles and Arcs

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  1. 10-6 Circles and Arcs

  2. Circles • In a plane, a circle is the set of all points equidistant from a given point called the center. • A diameter is a segment that contains the center and has both endpoints on the circle. • A radius is a segment that has one endpoint at the center and the other endpoint of the circle (congruent circles have congruent radii). • A central angle is an angle whose vertex is the center of the circle.

  3. Arcs • An arc is a part of a circle. • One type of arc, a semicircle, is half of a circle. • A minor arc is smaller than a semicircle. • Named by its endpoints • A major arc is larger than a semicircle. • Named by its endpoints and another point on the arc (so is a semicircle)

  4. Naming Arcs • What are the minor arcs of O? • What are the semicircles of O? • What are the major arcs of O?

  5. Arc Measure • The measure of a minor arc is equal to the measure of its corresponding central angle. • The measure of a major arc is the measure of the related minor arc subtracted from 360˚. • The measure of a semicircle is 180˚.

  6. Adjacent Arcs • Adjacent arcs are arcs of the same circle that have exactly one point in common. • You can add the measures of adjacent arcs to find the measure of the “larger” arc. Arc Addition Postulate: The measure of the arc formed by two (or more) adjacent arcs is the sum of the measures of the arcs.

  7. Finding the Measures of Arcs • What is the measure of each arc?

  8. Circumference • The circumference of a circle is the distance around the circle. • The number pi (π) is the ratio of the circumference of a circle to its diameter. Circumference of a Circle: C = πd = 2πr • Coplanar circles that have the same center are concentric circles.

  9. Finding a Distance • A car has a circular turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. What is the distance that the “outer” tire travels in one full turn? • How far does the “inner” tire travel in the same turn?

  10. Arc Length • The measure of an arc is in degrees while the arc length is a portion of a circle’s circumference (measured in cm, in., ft, etc.). Arc Length: The length of an arc of a circle is the product of the ratio of the arc measure to 360 and the circumference of the circle. • Congruent arcs are arcs that have the same measure AND are in the same or congruent circles.

  11. Finding Arc Length • What is the length of each red arc? Leave answers in terms of π.

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