6. 12,080 in. 2 7. 2475 in. 2 8. 1168.5 m 2 9. 2192.4 cm 2 10. 841.8 ft 2 11. 27.7 in. 2

6. 12,080 in.2

7. 2475 in.2

8. 1168.5 m2

9. 2192.4 cm2

10. 841.8 ft2

11. 27.7 in.2

12. 93.5 m2

13. 72 cm2

GEOMETRY LESSON 7-5GEOMETRY LESSON 7-5

Pages 382-385 Exercises

1. m 1 = 120; m 2 = 60; m 3 = 30

2. m 4 = 90; m 5 = 45; m 6 = 45

3. m 7 = 60; m 8 = 30; m 9 = 60

4. 2144.475 cm2

5. 2851.8 ft2

14. 384 3 in.2

15. 300 3 ft2

16. 162 3 m2

17. 75 3 m2

18. 12 3 in.2

19. a. 72

b. 54

20. a. 45

b. 67.5

21. a. 40

b. 70

22. a. 30

b. 75

23. 310.4 ft2

24. a. 9.1 in.

b. 6 in.

c. 3.7 in.


24. (continued)d. Answers may

vary. Sample: About 4 in.;

the length of a side of a pentagon

should be between 3.7

in. and 6 in.

25. m 1 = 36; m 2 = 18; m 3 = 72

26. The apothem is one leg of a rt. and the radius is the hypotenuse.

27. 73 cm2

28. 130 in.2

29. 27 m2

30. 103 ft2

31. 220 cm2

32. a–c.

regular octagon


32. (continued)d. Construct a 60° angle with vertex at circle’s center.

33. 600 3 m2

34. Check students’ work.

35. 128 cm2

36. 24 3 cm2, 41.6 cm2

37. 900 3 m2, 1558.8 m2

81 32

12

32

12

32

12

41. (continued)

b. apothem = ;

A = ap = (3s)

A = s2 3

s 36

12

12

s 36

38. 100 ft2

39. 16 3 in.2, 27.7 in.2

40. m2, 70.1 m2

41. a. b = s; h = s

A = bh

A = s • s

A = s2 3

7.6 Circles and Arcs

Objective:To find the measures of central angles and arcsTo find the circumference and arc length

Definitions

Circle – the set of all points equidistant from a given point called the center

Center of a Circle – the point from which all points are equidistant

Radius – a segment that has one endpoint at the center and the other endpoint on the circle

Definitions

Congruent Circles – circles that have congruent radii

Diameter – a segment that contains the center of a circle and has both endpoints on the circle

Central Angle – an angle whose vertex is the center of the circle

Definitions

Circumference – the distance around the circle

Pi (∏) – the ration of the circumference of a circle to its diameter

Examples

What if we are given a pie chart that represents data that have been collected?

How can we find the measure of the arc or the measure of the angle?

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


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.

Circles and Arcs

Some info to really help The measure of the arc is the same as the

measure of the central angle which creates that arc

1320

132

Arcs

A semicircle is half of a circle

TP

S

R

TP

S

R

TP

S

R

A minor arc is smaller than a semicircle A major arc is greater than a

semicircle

.


Circles and Arcs

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, ADE and AED.

Two semicircles in the diagram have

point A as an endpoint, ADB and AEB.

Arcs

Adjacent Arcs – arcs of the same circle that have exactly one point in common

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

Concentric Circles – circles that lie in the same plane and have the same center

Arc length – a fraction of a circle’s circumference

Postulate 7-1: Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

mABC = mAB + mBC

mDXM = 56 + 180 Substitute.

mDXM = 236 Simplify.

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 = 56 + 40 Substitute.

mXY = 96 Simplify.

Find mXY and mDXM in C. .

mDXM = mDX + mXWM Arc Addition Postulate


Circles and Arcs

Circumference

The circumference of a circle is the product of pi (∏) and the diameter.

C = ∏d or C = 2∏r

C = d Formula for the circumference of a circleC = (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 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.

Draw a diagram of the situation.

C 3.14(24) Use 3.14 to approximate .C 75.36 Simplify.


Circles and Arcs

7-6

Arc Length

The length of an arc of a circle is the product of the ratio

measure of the arc

360

and the circumference of the circle

Length of AB = mAB/360 *2∏r

length of ADB = • 2 (18) Substitute.210360

The length of ADB is 21 cm.

Find the length of ADB in M in terms of ..


Circles and Arcs

length of ADB = 21

7-6

mADB 360

length of ADB = • 2 r Arc Length Formula

Because mAB = 150,

mADB = 360 – 150 = 210. Arc Addition Postulate

1. A circle graph has a section marked “Potatoes: 28%.” What is the measure of the central angle of this section?

2. Explain how a major arc differs from a minor arc.

Use O for Exercises 3–6.

3. Find mYW.

4. Find mWXS.

5. Suppose that P has a diameter 2 in. greater than the diameter of O. How much greater is its circumference? Leave your answer in terms of .

6. Find the length of XY. Leave your answer in terms of .

.

..

A major arc is greater than a semicircle. A minor arc is smaller than a semicircle.

100.8

270

30

2

9


Circles and Arcs

Assignment

P. 389-390 #1-32 odd, 34-39

Documents

6. 12,080 in. 2 7. 2475 in. 2 8. 1168.5 m 2 9. 2192.4 cm 2 10. 841.8 ft 2 11. 27.7 in. 2