Math 3 Name ________________________ 5-1 Angles, Chords, and Arcs of Circles Learning Goals: I can identify/describe relationships among inscribed angles, arc measure, central angles, radii, chords and tangent lines. I can use knowledge of circles to solve problems Definitions: Arc – a connected section of the circumference of the circle (measured in degrees) Central Angle - an angle formed in the center of a circle by the meeting of two radii Chord - a straight line connecting two points on an arc or circle Diameter - a straight line running from one side of a circle through the center to the other side Radius - a straight line extending from the center of a circle or sphere to its edge Inscribed Angle – an angle in a circle whose vertex and endpoints all lie on the circle Given the definitions above, give the vocabulary word that describes the following: 1. BG 2. CF 3. ∠BGF 4. ∠ACF 5. DE 6. FB 7. Use the circles below to: a. Name the inscribed angle in circle B b. Name the central angle in circle O c. Use a protractor to find the measure of ABC ? What is the measure of ADC ? d. What is the measure of MON ? What is the measure of MPN ? e. If the measure of the central angle is known, how can you determine the measure of the inscribed angle? f. Inscribed Angle Theorem - In a circle, the measure of an inscribed angle is _______ the measure of its central angle. g. If you start at one point on a circle and rotate all the way around, how many degrees have your rotated? That is, how many degrees are in a circle? h. Given your answer to part (g), how many degrees are in one quarter of a circle? i. In circle C at the right, AB is one quarter of the circle. What is the measure of AB ? j. What is the measure of ACB ? g. What is the relationship between the central angle of a circle and the arc measure of the same circle? h. What is the relationship between an inscribed angle of a circle and the arc measure of the same circle? For problems 8-14, use circle C to the right to find the following measures, where C is the center of the circle. 8. m∠ACB 9. m∠ADB ̂ 10. m ̂ 11. m ̂ 12. m ̂ 13. m ̂ 14. m 15. In the circle to the right, D is the center and AC is the diameter. a. What is the measure of ADC ? b. What is the measure of ABC ? c. Angles that are inscribed on a diameter will always equal ________ Use the diagram at the right to answer questions 16. C is the center of the circle. ̂ = 112o, find m∠C and m∠D. 16. If Use the diagram at the right to answer questions 17-18. C is the center of the circle. Note that each problem is independent of the others! Do NOT refer to problem 17 in order to solve 18! ̂ = 12 + 14 and m∠C = 20x – 18. 17. Let ̂ and m∠C and m∠ADB Solve for x, ̂ = 20 + 4 and m∠D = 11x – 3. 18. Suppose ̂ m∠D, m∠ACB. Solve for x, 19. Draw a line segment from AB that bisects angle C. (call the intersection of your line segment and AB point M.) a. The line you drew creates two triangles that look congruent. Explain how you know that both of the triangles are congruent. b. Using the picture to the right, find the length of AB. c. If a diameter bisects a chord, what else do you know about the diameter and the chord? d. If a diameter bisects a chord, what do you know about the diameter and the central angle formed from the chord? 20. Using the picture to the right, find the length of the radius of the circle? So far we have seen that we can measure the degree of an arc on a circle. The degree measure tells us how far we rotate to form the arc. However, we can also measure the length of an arc of a circle. Arc length is the distance along the arc from one point to the next (as if you were walking along the arc). We find the length of the arc by using the circumference (distance around the circle) and the measure of the arc. EXAMPLE Find the length of minor arc AB Find the length of major arc ADB 21. Find the length of AB . 22. Find the length of ADB A sector of a circle is a “slice” of the circle (think a slice of pizza). Finding the area of the sector of a circle is very similar to finding the arc length, except we are using area instead of circumference. See below. EXAMPLE Find the area of sector ACB 23. Find the area of sector ACB 24. Mr. Grano ordered a pizza last night and ate a few pieces of it. If the central angle created by the missing pieces is 110o, and the radius of the pizza is 8 inches. Find the area of the pizza that Mr. Grano ate. A tangent line to a circle is a line that intersects a circle at only one point. The angle formed by the intersection of the radius of the circle and tangent line always forms a right angle. 25. Name two tangent lines in the diagram to the right. 26. Name two right angles in the diagram to the right. 27. Suppose a satellite is orbiting the Earth and is currently at point S. In its view of Earth in the plane of the equator, the angle between the lines of sight at S is 50o. The radius of Earth is 3,963 miles. a. What is the distance from S to the horizon along the equator (point T). b. How high is the satellite S above the Earth’s surface, that is, find the length of a segment S to the closest point on Earth’s surface along the equator?