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 chordC 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
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
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 isabout 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 m∠MCA = __ 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
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)º≈93.6 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 m∠COD = 90, COD is an isosceles right triangle. So CD = 25√2 cm, or about 35.36 cm.
c. Because m∠FOE = 120, m∠F = 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
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
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
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