angles and arcs
DESCRIPTION
Angles and Arcs. Recognize major arcs, minor arcs, semicircles, and central angles and their measures. Find arc length. An artist’s rendering of Larry Niven’s Ringworld. Two artist’s conceptions of views from the surface of Ringworld. ANGLES AND ARCS. - PowerPoint PPT PresentationTRANSCRIPT
Angles and Arcs• Recognize major arcs, minor arcs, semicircles, and central
angles and their measures.• Find arc length.
An artist’s rendering of Larry Niven’s Ringworld.
Two artist’s conceptions of views from the surface of Ringworld.
ANGLES AND ARCS
A central angle has the center of a circle as its vertex, and its sides contain two radii of the circle.
The sum of the measures of the angles around the center of a circle is 360°.
Key Concept Sum of Central Angles
The sum of the measures of the central angles of a circle with no interior points in common is 360°.
360321 mmm
1
23
Example 1 Measure of Central Angles
a) Find mAOB
25x°
3x°
2x°
OA
B
CD
E
b) Find mAOE
Key Concept Arcs of a CircleMinor Arc Major Arc Semicircle
110°
A
B C60°
DE
F
G
JK
LM
N
Usually named using the letters of the two endpoints.
Named by the letters of the two endpoints and another point on the arc.
Named by the letters of the two endpoints and another point on the arc.
AC DFE JML and JKL
Key Concept Arcs of a CircleMinor Arc Major Arc Semicircle
110°
A
B C60°
DE
F
G
JK
LM
N
Usually named using the letters of the two endpoints.
Named by the letters of the two endpoints and another point on the arc.
Named by the letters of the two endpoints and another point on the arc.
AC DFE JML and JKL
AC DFE JKL
JML
Theorem
In the same or in congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent.
Postulate Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
SQ
P
R
In circle S, mPQ + mQR = mPQR
Example 2 Measures of Arcs
FA B
CD
E
50°
a) Find mBE
b) Find mCBE
c) Find mACE
Example 3 Circle GraphsBICYCLESThis graph shows the percent of each type of bicycle sold in the United States in 2001.
Mountain 37% 133.2°
Youth 26% 93.6°
Hybrid 8% 28.8°
Other 8% 28.8°
Comfort 21% 75.6°
Identify any arcs that are congruent.
ARC LENGTH
Another way to measure an arc is by its length.
An arc is part of the circle, so the length of an arc is part of the circumference.
Example 4 Arc Length
P
R
120°
15
In circle P, PR = 15 and mQPR = 120°. Find the length of arc QR.
Solution:
r = 15, so C = 215 or 30, andarc QR = mQPR or 120°.
Write a proportion to compare each part to its whole.
l
l
l
10
)30(360120
30360120
l 31.42 units
Q
Key Concept Arc Length
rlA2360
degree measure of an arc degree measure of whole circle
arc length circumference
The proportion in the last example can be adapted to find an arc length in any circle.
This can also be expressed as
lCA
360