Download Circles: Arcs, Chords, and Angles - High School Geometry Notes and more Essays (high school) Mathematics in PDF only on Docsity!
Circles
Notes: Arcs and Central Angles
Circles have 3 types of arcs and their lengths are measured in degrees.
Minor Arcs Major Arcs Semi-Circle
Less than 180° Between 180° and 360° 180°
Name with 2 letter Name with 3 letters Name with 3 letters
AB CED , EDC , CDE , ECD , DCE , DEC FGH
Central Angles
- Angles with their vertex located at the center of the circle.
- Central angles have the same measure as the arcs they intercept.
Find the value of the missing variables.
A
B
C
D
E
F
G
H
O
x
x
y
x
x
x
x
y
Theorems involving chords of a circle:
- If a line through the center of a circle is perpendicular to a chord, then 3 facts are true:
Examples: Draw a picture, then solve.
- Suppose a chord of a circle is 24 cm long and is 9 cm from the center of the circle. Find the length of the radius.
- The radius of a circle is 15 cm. Find the length of a chord 12 cm from the center of the circle.
- The diameter of a circle is 10 in long and the length of a chord is 6 in. Find the distance from the chord to the center of the circle.
- Find the length of a chord of a circle of radius 13cm if its distance from the center of the circle is 5 cm.
- Find the length of the radius of a circle if a 14 inch chord is 24 inches from the center of the circle.
- How far from the center of a circle of radius 10 mm is a chord of length 16 mm?
- A chord of a circle is 10 in long and 12 in from the center of the circle. Find the length of the diameter of the circle.
1.____________________________________________________
2.___________________________________________________
3.___________________________________________________
More on types of Angles(inside and outside)
INSIDE ANGLES: ( NOT INSCRIBED, NOT CENTRAL!) An angle inside the circle formed by 2 intersecting
chords.
m(Inside ) =
arc + arc
Ex.
m 1 = ____ m 2 = ____
m 3 = _____ m4 = ____
Ex. 2
X = ________
Ex. 3
Ex. 4
x
x
y
x
OUTSIDE ANGLES: Angles formed by segments outside of a circle
M (outside angle) =
outer arc − inner arc
CASE 1 : 2 SECANT SEGMENTS CASE 2: 2 TANGENTS CASE 3: 1 SECANT/ 1 TANGENT
X = ______ x = _____ x = _____
Find the numbered angle measures or “x”.
x 20 70
x
x
x
Review
Solve.
- Find the length of a chord of a circle of radius 13cm if its distance from the center of the circle is 5 cm.
- Find the length of the radius of a circle if a 14 inch chord is 24 inches from the center of the circle.
- How far from the center of a circle of radius 10 mm is a chord of length 16 mm?
- A chord of a circle is 10 inches long and 12 inches from the center of the circle. Find the length of the
diameter of the circle.
Find the missing angle or arc.
a = ________
b = ________
c = ________
a = ________
b = ________
c = ________
a = ________
b = ________
c = ________
a = ________
b = ________
c = ________
a = ________
b = ________
c = ________
a = ________
b = ________
c = ________
˚
4 ˚
b˚ a˚
c˚
˚ ˚
9 ˚
b˚
a˚
c˚
b˚
a˚
c˚
b˚
˚
c˚
a˚
˚
˚
˚
8 ˚
c˚
a˚
b˚
a˚
˚
b˚
c˚
8 ˚
A B C D
● x
x
x
x
y
y
x
x
x
y
x
●
3y
y
x
●
x
●
x
x
Area of Circles
Area of a Circle: A=
Ex 1: Find the area of the following circles.
(a) (b) (c)
Ex 2: Given the following areas, find the radius or diameter of each circle.
(a) A = 64π (b) A = 25π
Ex 3: Find the area of the shaded region in each figure.
(a) (b)
(c) (d)
Area of Rhombus:
Circles
NOTES ON SECTOR OF A CIRCLE
SECTOR: A region bounded by 2 radii and an arc. (slice of pizza)
area of a sector is “part” of the area
--- or ---
Ex’s: Find the area of the shaded region:
60
6 cm
12 cm •
60
9 cm
