How to find the measures of central angles and arcs, and to find circumference and arc length. Chapter 10.6GeometryStandard/Goal 2.2, 4.1

1. Check and discuss the assignment from Friday. 2. Work on Quiz Read, write, and discuss how to find the measures of central angles and arcs. 4. Read, write, and discuss how to find circumference and arc length. 5. Work on assignment.

Circle is the set of all points in a plane that are equidistant from a given point, called _______ of the circle. A circle with center P is called “circle P ”, or Ο P center

Radius is a segment that has one endpoint at the center and the other endpoint on the circle. Congruent circles have the same radii. Diameter is a segment that contains the center of a circle and has both endpoints on the circle.

radius diameter center C P A B

Central angle of the circle is an angle whose vertex is the center of the circle. C P A B

Semicircle is half of a circle. EH G F 180˚

Minor arc of the circle is smaller than a semicircle. Major arc of the circle is greater than a semicircle.

Minor arc Major arc Central angle P C A B Naming Arcs

Measure of a Minor Arc is the measure of its central angle. 60˚ EH G F

Measure of a Major Arc is defined as the difference between 360˚ and the measure of its associated minor arc. 60˚ EH G F

Adjacent arcs are arcs of the same circle that have exactly one point in common.

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. A B C

Because there are 360° in a circle, multiply each percent by 360 to find the measure of each central angle. 65+ : 25% of 360 = = 90 45–64: 40% of 260 = = –44: 27% of 360 = = 97.2 Under 25: 8% of 360 = = 28.8 Lesson 10-6 A researcher surveyed 2000 members of a club to find their ages. The graph shows the survey results. Find the measure of each central angle in the circle graph.

. Lesson 10-6 Identify the minor arcs, major arcs, and semicircles in P with point A as an endpoint. Minor arcs are smaller than semicircles. Two minor arcs in the diagram have point A as an endpoint, AD and AE. Major arcs are larger than semicircles. Two major arcs in the diagram have point A as an endpoint, A DE and AED. Two semicircles in the diagram have point A as an endpoint, ADB and AEB.

mDXM = Substitute. mDXM = 236Simplify. mXY = mXD + mDY Arc Addition Postulate mXY = m XCD + mDY The measure of a minor arc is the measure of its corresponding central angle. mXY = Substitute. mXY = 96Simplify. Find mXY and mDXM in C.. mDXM = mDX + mXWM Arc Addition Postulate Lesson 10-6

Circumference of a circle is the distance around the circle.

The circumference C of a circle is: Where d is the diameter and r is the radius of the circle. d = diameter r = radius

Arc Length fraction of the circle’s circumference. You can use the measure of the arc (in degrees) to find its length (in linear units)

In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360˚. B A P Arc Length of Circumference

Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles.

C = d Formula for the circumference of a circle C = (24)Substitute. A circular swimming pool with a 16-ft diameter will be enclosed in a circular fence 4 ft from the pool. What length of fencing material is needed? Round your answer to the next whole number. The pool and the fence are concentric circles. The diameter of the pool is 16 ft, so the diameter of the fence is = 24 ft. Use the formula for the circumference of a circle to find the length of fencing material needed. About 76 ft of fencing material is needed. Draw a diagram of the situation. C 3.14(24)Use 3.14 to approximate. C 75.36Simplify. Lesson 10-6

length of ADB = 2 (18)Substitute The length of ADB is 21 cm. Find the length of ADB in M in terms of.. Lesson 10-6 length of ADB = 21 mADB 360 length of ADB = 2 r Arc Length Formula Because mAB = 150, mADB = 360 – 150 = 210.Arc Addition Postulate

Kennedy, D., Charles, R., Hall, B., Bass, L., Johnson, A. (2009) Geometry Prentice Hall Mathematics. Power Point made by: Robert Orloski Jerome High School.