Chapter 9: Circles Guided Notes
Circles GUIDED NOTES SUMMIT PUBLIC SCHOOLS CREATED BY: JAMES LAJOIE
Part One: Basic Terms & Tangents 2
Example 1: A) A radius has length 20x. Find its diameter. B) A diameter has length 31m. Find its radius.
Some New Terms! 3
Putting it all together! 4
Ex. 2 Consider the diagram below. Label the following (on the right)
as either a: Radius, Diameter, Secant, Chord, Tangent
Spheres 5
Definition: a collection of points in space that are
the same distance from a point (called the center) Example 3: a) Name points on the sphere. b) Name three radii. c) Name a diameter. d) Name a chord. e) Name a secant. f) Name a tangent. g) Name a point of tangency.
And more of the new… 6
Congruent Circles / Concentric Spheres 7
Congruent circles are:
circles that have congruent radii.
Concentric spheres are: spheres that have the same center.
Let’s go off on a tangent… 8
Example 4: Respond to the questions regarding the diagram.
Naming a circle: We name a circle by referring to its center. The circles in the coordinate plane above are circles A and B.
Inscribed vs. Circumscribed 9
Inscribed or Circumscribed? 10
Example 5: Tell whether the circle is being
inscribed or circumscribed.
Two Special Theorems 11
Point of Tangency Theorem 12
Example 6 13
Is AB tangent to circle C?
Example 7 14
Put on your thinking cap… 15
Example 8 16
Find the value(s) of x,
given that AB and AD are tangents to circle C.
Example 9 17
The circles in the diagram below are concentric, with
radii of 8 and 17. If JL is tangent to the circle of radius 8, find its length.
Example 10 18
In the diagram, R and S are points of tangency for circles Q and P, respectively. Find the length of RS.
Part Two: Arcs & Chords 19
Key Term: CENTRAL ANGLE A central angle of a circle is an angle formed with the following properties: 1) Its vertex is the center of the circle 2) Its sides are two radii of the circle Notice how the central angle in the diagram forms a “segment” on the circle. This is called an arc.
Major and Minor Arcs and Semicircles 20
Major and Minor Arcs 21
Arc Addition Post. & Congruent Minor Arcs Thm. 22
Congruent Minor Arcs Theorem: In the same circle or in congruent circles, two minor arcs are congruent if and only if…
Example 11 23
If the measure of the central angle is 85 degrees,
find:
mAC mABC
Example 12 24
In the three diagrams above, find the measure of the indicated arc. In the diagram to the right, find the measure of angle PCQ.
Example 13 25
Example 14 26
Example 15 27
Find all the possible values of x.
Theorems about Chords 28
Example 16 29
Find the length of XY.
Find the measure of arc MN.
Example 17 30
If the measure of arc AC is 150˚, find the measure of
arc AB.
More Theorems… 31
Example 18 32
Find x in the diagram to the right.
More Theorems… 33
Example 19 34
Find x in the diagram below.
Example 20 35
Use trigonometry to find the measure of the arc cut
off by a chord 14 inches long in a circle of radius 8 inches.
Part Three: Inscribed & Other Angles of Circles 36
Inscribed Angle Theorem 37
Example 21 38
Same Intercepted Arc Theorem 39
Example 22 40
Examples 23-24 41
Example 25 42
Find x and y.
Angle formed by a tangent and a chord… 43
Example 26 44
Find the measure of angle 1 or x in each diagram.
Example 27 45
Another angle theorem! 46
PROOF!
Example 28 48
Exterior Angles of Circles 49
PROOF!
Example 29 51
Find angle 1.
Find x in each diagram.
Example 30 52
Find x.
Example 31 53
Part Four: Segments of Chords 54
Example 32 55
Example 33 56
Find the value(s) of x.
External Secant Theorem 57
TRICK: Think… (WHOLE)(OUTER) = (WHOLE)(OUTER)
External Secant & Tangent Theorem 58
TRICK: Think…
(OUTER)2 = (WHOLE)(OUTER)
Example 34 59
Find x in each diagram.
Example 35 60
Find x in each diagram.
Part One: Basic Terms & Tangents 2
Example 1: A) A radius has length 20x. Find its diameter. B) A diameter has length 31m. Find its radius.
Some New Terms! 3
Putting it all together! 4
Ex. 2 Consider the diagram below. Label the following (on the right)
as either a: Radius, Diameter, Secant, Chord, Tangent
Spheres 5
Definition: a collection of points in space that are
the same distance from a point (called the center) Example 3: a) Name points on the sphere. b) Name three radii. c) Name a diameter. d) Name a chord. e) Name a secant. f) Name a tangent. g) Name a point of tangency.
And more of the new… 6
Congruent Circles / Concentric Spheres 7
Congruent circles are:
circles that have congruent radii.
Concentric spheres are: spheres that have the same center.
Let’s go off on a tangent… 8
Example 4: Respond to the questions regarding the diagram.
Naming a circle: We name a circle by referring to its center. The circles in the coordinate plane above are circles A and B.
Inscribed vs. Circumscribed 9
Inscribed or Circumscribed? 10
Example 5: Tell whether the circle is being
inscribed or circumscribed.
Two Special Theorems 11
Point of Tangency Theorem 12
Example 6 13
Is AB tangent to circle C?
Example 7 14
Put on your thinking cap… 15
Example 8 16
Find the value(s) of x,
given that AB and AD are tangents to circle C.
Example 9 17
The circles in the diagram below are concentric, with
radii of 8 and 17. If JL is tangent to the circle of radius 8, find its length.
Example 10 18
In the diagram, R and S are points of tangency for circles Q and P, respectively. Find the length of RS.
Part Two: Arcs & Chords 19
Key Term: CENTRAL ANGLE A central angle of a circle is an angle formed with the following properties: 1) Its vertex is the center of the circle 2) Its sides are two radii of the circle Notice how the central angle in the diagram forms a “segment” on the circle. This is called an arc.
Major and Minor Arcs and Semicircles 20
Major and Minor Arcs 21
Arc Addition Post. & Congruent Minor Arcs Thm. 22
Congruent Minor Arcs Theorem: In the same circle or in congruent circles, two minor arcs are congruent if and only if…
Example 11 23
If the measure of the central angle is 85 degrees,
find:
mAC mABC
Example 12 24
In the three diagrams above, find the measure of the indicated arc. In the diagram to the right, find the measure of angle PCQ.
Example 13 25
Example 14 26
Example 15 27
Find all the possible values of x.
Theorems about Chords 28
Example 16 29
Find the length of XY.
Find the measure of arc MN.
Example 17 30
If the measure of arc AC is 150˚, find the measure of
arc AB.
More Theorems… 31
Example 18 32
Find x in the diagram to the right.
More Theorems… 33
Example 19 34
Find x in the diagram below.
Example 20 35
Use trigonometry to find the measure of the arc cut
off by a chord 14 inches long in a circle of radius 8 inches.
Part Three: Inscribed & Other Angles of Circles 36
Inscribed Angle Theorem 37
Example 21 38
Same Intercepted Arc Theorem 39
Example 22 40
Examples 23-24 41
Example 25 42
Find x and y.
Angle formed by a tangent and a chord… 43
Example 26 44
Find the measure of angle 1 or x in each diagram.
Example 27 45
Another angle theorem! 46
PROOF!
Example 28 48
Exterior Angles of Circles 49
PROOF!
Example 29 51
Find angle 1.
Find x in each diagram.
Example 30 52
Find x.
Example 31 53
Part Four: Segments of Chords 54
Example 32 55
Example 33 56
Find the value(s) of x.
External Secant Theorem 57
TRICK: Think… (WHOLE)(OUTER) = (WHOLE)(OUTER)
External Secant & Tangent Theorem 58
TRICK: Think…
(OUTER)2 = (WHOLE)(OUTER)
Example 34 59
Find x in each diagram.
Example 35 60
Find x in each diagram.
