Math 2 Name __________________________Lesson 6-5: Angles, Chords, Arcs, Tangents, and Sectors 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)

Three types: Semicircle, Minor Arc, Major ArcChord - a line segment connecting two points on an arc or circleDiameter - a chord containing the center of the circleRadius - a line segment connecting the center of a circle to a point on the circleCentral Angle - an angle formed in the center of a circle by the meeting of two radiiInscribed Angle – an angle in a circle whose vertex and endpoints all lie on the circle

Based on the definitions above, give the vocabulary word that describes the following:

1. 2.

3. ∠BGF 4. ∠ACF

5. 6.

7. 8.

9. Use the circles below to:a. Name the inscribed angle in circle B. b. Name the central angle in circle O.

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2c. The following are angle measures from Circle B and Circle O:

, , ,

Based on your measurements, 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 the central angle with the same endpoints.

10. 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?

a. How many degrees are in one quarter of a circle?

b. In circle C at the right, is one quarter of the circle.

What is the measure (in degrees) of ?

c. What is the measure of ?

d. What is the relationship between the central angle of a circle and the measure (in degrees) of an arc with the same endpoints?

e. What is the relationship between an inscribed angle of a circle and the measure of an arc with the same endpoints?

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For problems 11 ˗ 18, use circle C to the right to find the following measures, where C is the center of the circle.

11. m∠ACB =

12. m∠ADB =

13.

14.

15.

16.

17.

18.

19. In the circle to the right, D is the center and is the diameter. a. What is the measure of Explain how you know.

b. What is the measure of Explain how you know.

c. Angles that are inscribed on a diameter will always equal ________.

d. Angles that are inscribed in a semicircle will always equal ________.

e. If the radius of circle D is 3 cm and , find the length of

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4 Use the diagram at the right to answer question 20. C is the center of the circle.

20. If , find m∠ACB and m∠ADB. Explain how you know.

Use the diagram at the right to answer questions 21 & 22. C is the center of the circle. Note that each problem is independent of the others! Do NOT refer to problem 21 in order to solve 22!

21. Let and m∠ACB = 20x – 18. Solve for x, m AB and m∠ACB and m∠ADB

22. Suppose and m∠ADB = 11x – 3.

Solve for x, m∠ADB, m∠ACB.

A

B

Z

C

B

A

D

C

Page 523. .

= 10x + 7Find

24. .

.Find

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 along the circle 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.

To understand the difference between the degree measure of an arc and the length of an arc, see the figure to

the right. Since is the same angle as , then both and have a degree measure of

because that is the measure of the central angle. However, if you think about “walking” along arc

versus walking along , clearly the length of is greater.

Hence, the measure of an arc does NOT depend on the size of the circle, but the length of the arc DOES depend on the size of the circle.

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Find the length of minor arc .

Find the length of major arc .

25. Find the length of . 26. Find the length of .

page 7A 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.

EXAMPLEFind the area of sector ACB.

28. Find the area of sector ACB.

28. 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.

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29. Pac Man’s mouth is open at a 90◦ angle. Find the area of Pac Man if the radius (edge of his mouth) is equal to 22 mm.

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.

30. Name two tangent lines in the diagram to the right.

31. Name two right angles in the diagram to the right.

Page 932. 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 The radius of Earth is 3,963 miles.

a. What is the distance from S to the horizon along the equator (point T).

Note that if you draw the line segment it will create two congruent triangles.

EXTENSION: 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?

OVER SAT/ACT Practice

33.

34.

35.