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14-1. Chord Length and Arc Measure. Vocabulary. Congruent Circles and Congruent Arcs. Lesson. Mental Math

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824 Further Work with Circles

Lesson

Chord Length and

Arc Measure

14-1

BIG IDEA Properties of chords in circles follow from

properties of isosceles triangles. Knowing any two of the length of a chord, the measure of its arc, and the radius of the circle, you can determine the third.

The Flower Fountain in Centennial Park in Holland, Michigan, is protected by an iron fence that is in the shape of a regular 16-gon. Recall that the vertices of a regular polygon all lie on a single circle. The sides of that polygon are chords of the circle. In general, a segment

____

DB is called the

chord of arc DB . If the center of the 

circle is C, then ∠BCD is called the

central angle of the chord

____

DB .

Congruent Circles and Congruent Arcs

If two circles X and Y have equal radii, then X can be mapped onto Y by the translation vector XY. So X is congruent to Y.

If two circles do not have equal radii, no isometry will map one onto the other, since isometries preserve distance. These arguments prove that circles are

congruent if and only if their radii have the same length.

−−

BE is a diameter of circle C.

a. What kind of angle

is ∠ACB?

b. What kind of angle

is ∠EBD?

c. How does m∠ACB compare to m AB ? 

d. How does m∠EBD compare to m ED ? 

e. Which of AB and  BDA is the minor arc? Which the major arc?

Mental Math

Vocabulary

chord of an arc central angle of a chord

C A B E D Y X C B D central angle of DB chord of DB SMP_SEGEO_C14L01_824-830.indd 824 SMP_SEGEO_C14L01_824-830.indd 824 5/27/08 10:26:36 AM5/27/08 10:26:36 AM

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Chord Length and Arc Measure 825

Suppose two arcs AB and  CD have the same measure, as in the 

circle at the right. Then you can rotate AB about O by the measure 

of ∠AOC to the position of CD . Then the chord 

___ AB rotates to ____ CD , so ___ AB  ____

CD . Thus, in a circle, arcs of the same measure are congruent

and have congruent chords. This proves Part a of the next theorem. The proof of Part b is left for you as Question 17.

Arc-Chord Congruence Theorem

In a circle or in congruent circles:

If two arcs have the same measure, they are congruent and their chords are congruent.

If two chords have the same length, their minor arcs have the same measure.

a. b.

The Arc-Chord Congruence Theorem applies only in a single circle or in circles with the same radii. In circles with different radii, arcs of the same measure cannot be congruent, nor can their chords be congruent. Even so, in those circles, arcs with the same measure are

similar, because one can be mapped onto the other by a composite

of a translation, a size change, and a rotation. All circular arcs with

the same measure are similar.

Properties of Chords

When an arc of a circle is not a semicircle, the radii drawn to the endpoints of the arc and the arc’s corresponding chord form an isosceles triangle. Recall from Lesson 6-2 that, in an isosceles triangle, the bisector of the vertex angle, the perpendicular bisector of the base, the altitude from the vertex, and the median from the vertex all lie on the same line. In the language of circles and chords, this leads to the following theorem.

Chord-Center Theorem

The line containing the center of a circle perpendicular to a chord bisects the chord.

The line containing the center of a circle and the midpoint of a chord bisects the central angle of the chord.

The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord.

The perpendicular bisector of a chord of a circle contains the center of the circle.

1. 2. 3. 4. B C D O A B O A SMP_SEGEO_C14L01_824-830.indd 825 SMP_SEGEO_C14L01_824-830.indd 825 5/27/08 10:26:40 AM5/27/08 10:26:40 AM

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826 Further Work with Circles

As you know, the distance from a point to a line is the length of the perpendicular segment from the point to the line. By applying the Chord-Center Theorem and the Pythagorean Theorem, if you know any two of the radius of the circle, the length of a chord, or the distance of the chord from the center of the circle, you can fi nd the third.

Example 1

A chord 13 cm long lies in a circle of radius 10 cm. How far is the chord from the center of the circle?

Solution Draw a diagram with the given information: a circle with radius 10 cm and a chord that is 13 cm in length. Draw the full isosceles triangle because it helps in thinking about the question. The goal is to fi nd the distance OL from the center O to the chord.

Because

__

LO ⊥

___

NM , by Part (1) of the Chord-Center Theorem,

__

LO bisects

___

MN .

This means ML = LN = 6.5. You now know the leg and the hypotenuse of a right triangle. By the Pythagorean Theorem, 6.52+ OL2= 102, from

which OL2= 57.75, and so OL =

_____57.75 7.59934. The chord is

about 7.60 cm from the center of the circle.

How to Find the Length of a Chord

Given the measure of an arc and the radius of the circle, the length of its corresponding chord can always be found using trigonometry. Now we return to the iron fence around the Flower Fountain.

Example 2

Colleen would like to hang a string of lights on the fence around the Flower Fountain. The fence is a 16-gon inscribed in a circle of radius 15 ft. How long a string of lights will she need to go once around the fountain?

Solution A picture is drawn at the right. Chord

__

AB is one side of the

fence. Multiplying this length by 16 yields the total length of the string of lights needed.

ACB is isosceles with vertex angle C. AB is  __161 of the circle, so m AB  = m∠ACB = ____16? º

Let M be the midpoint of ___

AB . Then, by Parts 2 and 3 of the Chord-Center Theorem,

___

CM bisects ? . Also, ? is a right triangle. So mMCA = __ 12 · _____ ?16º = 11.25º.

To fi nd AM, use trigonometry.

M O 10 10 N L GUIDED M C A B SMP_SEGEO_C14L01_824-830.indd 826 SMP_SEGEO_C14L01_824-830.indd 826 5/27/08 10:26:43 AM5/27/08 10:26:43 AM

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Chord Length and Arc Measure 827 sin(11.25)º ____AM

?

AM ? · sin(11.25)º

Because AM=__12AB, AB=2 · AM 2 · ? = ? . She needs 16· ? = 480 · sin(11.25)º93.6 ft of rope.

If the measure of an arc is a multiple of 30 or 45, it is possible to fi nd the exact length using special right triangles and without trigonometry.

Example 3

In a circle with a radius of 25 centimeters, fi nd the length of a chord of an arc with the given measure.

a. 60º b. 90º c. 120º

Solution Always draw a picture.

a. AOB is an isosceles triangle with a 60º vertex angle, so it is equilateral. Therefore, AB = 25 cm.

b. Because mCOD = 90, COD is an isosceles right triangle. So CD = 25√2 cm, or about 35.36 cm.

c. Because mFOE = 120, mF = 30 and m___ E = 30. Now, draw the altitude

OG to the base of OEF. The two triangles formed are 30-60-90 triangles.

OE = 2 · OG, so OG =___252 cm. GE = √3 · OG, so GE =___252 √3 cm. Therefore, FE = 2 · GE = 25√3 cm 43.30 cm. QY B O 60° 60° 25 cm 25 cm A D O 90° 25 cm 25 cm C E O 60° 60° 120° 25 cm 25 cm F G QY

What is the perimeter of an equilateral triangle whose vertices are on a circle with radius 25 in.?

SMP_SEGEO_C14L01_824-830.indd 827

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828 Further Work with Circles

Questions

COVERING THE IDEAS

In 1−4, use the diagram at the right.

1. Name three chords.

2. True or False If m CB = m  CD , then 

___

CB 

____

CD .

3. True or False If m CB = m  CD , then m  FE = m  CD .

4. m CB = m  FE , yet 

___

CB is not congruent to

___

FE . Explain why.

5. Multiple Choice Two circles are congruent if and only if

A they have the same center.

B they have radii equal in length.

C they have arcs of the same length.

D they have chords of the same length.

Fill in the Blanks In 6−9, use the diagram at the right.

6. If ____ GK

___HI and ____ HK = 6 cm, then HI = ? . 7. If K is the midpoint of ___

HI and m∠HGI = 140, then

m∠HGK = ? .

8. If GK bisects ∠HGI and HI = 6 cm, then HK =  ? and

m∠GKH = ? .

9. Explain why GHI is isosceles.

10. A 30-cm chord lies in a circle of radius 17 cm. Find the distance from the center of the circle to the chord.

In 11−14, O at the right has radius 10 m.

11. Find the exact length of a chord of a 60º arc. 12. Find the exact length of a chord of a 90º arc. 13. Find the exact length of a chord of a 120º arc.

14. Find the approximate length of a chord of a 23º arc to the nearest thousandth.

15. Refer to Example 2. How far off would Colleen have been in her calculation if, to estimate the length of lights needed, she calculated the circumference of the circle rather than the perimeter of the regular polygon?

16. Find the perimeter of a regular 20-gon that is inscribed in a circle of radius 1. F E D C A B K I H G O 10 m SMP_SEGEO_C14L01_824-830.indd 828 SMP_SEGEO_C14L01_824-830.indd 828 5/27/08 10:26:50 AM5/27/08 10:26:50 AM

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Chord Length and Arc Measure 829

APPLYING THE MATHEMATICS

17. Complete this proof of Part b of the Arc-Chord Congruence Theorem. Given AB = CD

Prove m AB  = m CD 

(Hint: The measure of an arc equals the measure of its central angle.)

18. Of all the planets, the orbit of Venus is closest to being a circle. Venus makes one full revolution around our Sun every 225 days at a distance of about 67.2 million miles from the Sun. From one Earth day to the next, how far has Venus traveled?

19. A circle of radius 10 cm lies on a sphere. Its center is 5 cm from the center of the sphere. Find the radius of the sphere.

20. Find the exact perimeter of an equilateral triangle that is inscribed in the circle with equation x2 + y2 = 100. 21. To allow access to the sewer level,

workmen must climb down a manhole. A workman is concerned about fi tting down the hole. If a ladder of width 40 cm is built into a manhole of diameter 61 cm, then what is the maximum distance from the ladder to the edge of the manhole?

22. Mariana found one of her grandmother’s antique plates. The plate is lined with gold border in the shape of a regular decagon. The decagon’s vertices are on the edge of the plate. If the diameter of the plate is 8", fi nd the length of the gold border to the nearest thousandth.

REVIEW

23. The fi gure below shows a stepladder. If the slant height of the ladder is 13', what height is the top rung above the ground? (Lesson 13-6) 13' 65° O D C B A

It takes 243 Earth days for Venus to rotate on its axis once, longer than it takes for Venus to orbit the Sun.

40 cm

ladder

SMP_SEGEO_C14L01_824-830.indd 829

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830 Further Work with Circles

24. Given that ABCD is a rectangle, fi nd m∠DBA to the nearest tenth of a degree. (Lesson 13-5)

25. A curtain is placed inside a circular auditorium. When viewed from the center of the auditorium, it takes up 64.7º of the viewer’s fi eld of vision, as shown in the fi gure below. What degree of the viewer’s fi eld of vision does it take up when viewed from points L, M, and N ? (Lesson 6-3)

M L

N

64.7˚

26. a. Trace the triangle below, and construct a point that is equidistant from all three of its vertices.

b. How does the point from Part a relate to the circle in which

the triangle can be inscribed? (Lesson 5-5)

EXPLORATION

27. Use a DGS. Construct a circle with radius 1. Let

___

AC be a

diameter of the circle. Place point B on the circle. Move point B from point A to point C. As point B moves, record x = m AB and  y = length of the chord

___

AB . Graph the points (x, y) and describe

the graph. QY ANSWER 75 √  3 ≈ 129.9 in. 9x 15x A B C D SMP_SEGEO_C14L01_824-830.indd 830 SMP_SEGEO_C14L01_824-830.indd 830 5/27/08 10:26:56 AM5/27/08 10:26:56 AM

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