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Objectives: Find measures of central angles and arcs Find circumference and arc length

Section 7-6 Circles and Arcs SPI 32B: Identify central angles of circles given a diagram SPI 33A: Solve problems involving the properties of arcs. Objectives: Find measures of central angles and arcs Find circumference and arc length.

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Objectives: Find measures of central angles and arcs Find circumference and arc length

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  1. Section 7-6 Circles and Arcs SPI 32B: Identify central angles of circles given a diagramSPI 33A: Solve problems involving the properties of arcs • Objectives: • Find measures of central angles and arcs • Find circumference and arc length Geometric Vocabulary Of a Circle

  2. Vocabulary of a Circle • Circle: • set of all points equidistant from the center • named by its center (⓪P) • Radius: • segment that has one endpoint at the center and the other on the circle ( ) • Diameter: • segment that contains the center of the circle and has both endpoints on the circle ( is the diameter)

  3. More Vocabulary • Congruent circles: • circles that have congruent radii • Central Angle: • angle whose vertex is the center of the circle • CPA is a central angle

  4. Circles and Arcs 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. Because there are 360° in a circle, multiply each percent by 360 to find the measure of each central angle. 65+ : 25% of 360 = 0.25 • 360 = 90 45–64: 40% of 260 = 0.4 • 360 = 144 25–44: 27% of 360 = 0.27 • 360 = 97.2 Under 25: 8% of 360 = 0.08 • 360 = 28.8

  5. and yet…More Vocabulary • Arc: • is part of a circle (part of the circumference) • denoted by the ( arc symbol) 3 Types of Arcs: Major Arc: greater than a semicircle (greater than 180º) Minor arc: smaller than a semicircle (less than 180º Semicircle: Half a circle 180º

  6. Arc Addition Postulate • Adjacent Arcs • Arcs of the same circle that have exactly one point in common. • Measures of adjacent arcs can be added like adjacent angles

  7. . 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, ADE and AED. Two semicircles in the diagram have point A as an endpoint, ADB and AEB. Circles and Arcs Identify the minor arcs, major arcs, and semicircles in P with point A as an endpoint.

  8. . mXY = mXD + mDYArc Addition Postulate mXY = m XCD + mDYThe measure of a minor arc is the measure of its corresponding central angle. mXY = 56 + 40 Substitute. mXY = 96 Simplify. mDXM = mDX + mXWMArc Addition Postulate mDXM = 56 + 180 Substitute. mDXM = 236 Simplify. Circles and Arcs Find mXY and mDXM in C.

  9. Circumference and Arc Length Circumference is the distance around a circle.

  10. Draw a diagram of the situation. C = dFormula for the circumference of a circle C = (24) Substitute. C 3.14(24) Use 3.14 to approximate . C 75.36 Simplify. Circles and ArcsConcentric circles have the same center. 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 16 + 4 + 4 = 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.

  11. . Because mAB = 150, mADB = 360 – 150 = 210. Arc Addition Postulate 210 360 length of ADB = • 2 (18) Substitute. mADB 360 length of ADB = • 2 rArc Length Formula The length of ADB is 21 cm. length of ADB = 21 Circles and Arcs Find the length of ADB in M in terms of .

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