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Circles – Basic Terms
A circle is the set of points in
a plane at a given distance
from a given point in that
plane.
The center is the given point.
The radius is the given
distance. All radii of a circle
are congruent.
Circles – Basic Terms
A chord is a segment whose
endpoints lie on a circle.
A secant is a line that
contains a chord.
A diameter is a chord that
contains the center of the
circle.
Circles – Basic Terms
A sphere with center O and
radius r is the set of all points
in space at a distance r
from point O. Many terms
used with circles are also
used with spheres:
Radii
Chord
Tangent
Diameter
Secant
Center
Circles – Basic Terms
Congruent circles (or
spheres) are circles (or
spheres) that have
congruent radii.
Circles – Basic Terms
Concentric Circles are
circles that lie in the same
plane and have the same
center.
Concentric Spheres are
spheres that have the same
center.
Circles – Basic Terms
A polygons is inscribed in a
circle and the circle is
circumscribed about the
polygon when each vertex
of the polygon lies on the
circle.
Polygon ABCD inscribed in a
circle.
A circle circumscribed about
polygon ABCD.
Learning Log Summary
LT 1 - I can apply theorems that relate to
tangents and radii in proof and problem
solving.
Key vocabulary for circles includes…
A tangent line is…
Proof by Contradiction
“When you have eliminated all which is
impossible, then whatever remains, however
improbable, must be the truth.”
-Sir Arthur Conan Doyle
Proof by Contradiction
Trying to prove: If p, then q.
Assume: “not q”
Show: this assumption leads to a contradiction, so
the original conclusion must be true.
Proof by Contradiction
1. Prove that if m and n are odd integers, then
𝑚 ∙ 𝑛 is also odd.
Proof by Contradiction
Given: ∠𝐴 ≅ ∠𝐷 , AB=DE , AC ≠ DF
Prove: ∠𝐵 ≠ ∠𝐸
Proof by Contradiction
Given: P is not the mdpt of HK ; HJ=JK
Prove: JP does not bisect ∠HJK
Learning Log Summary
LT 2 - I can use indirect proof to prove an
algebraic or geometric statement.
The steps to indirect proof are…
The goal in a proof by contradiction is…
Theorem 9-1
Theorem 9-1: If a line is tangent to a circle, then the line is
perpendicular to the radius drawn to the point of tangency.
Given: 𝑙 is tangent to ⊙ 𝐶 at A
Prove: 𝐶𝐴 ⊥ 𝑙
http://bit.ly/2kyqbQk
Theorem 9-1
Theorem 9-1: If a line is tangent to a circle, then the line is
perpendicular to the radius drawn to the point of tangency.
Given: 𝑙 is tangent to ⊙ 𝐶 at A
Prove: 𝐶𝐴 ⊥ 𝑙
PARTNER A
• Describe the assumption
you will make.
PARTNER B
• Explain why your partner is
making this assumption.
• Re-state each step of your
partner’s reasoning, asking
clarifying questions.
• Explain each step of your
reasoning, including the
justification for that step.
• Explain clearly where the
contradiction is!
• What is the conclusion that
can be drawn?
Corollary
Corollary: Tangents to a circle from a point are
congruent.
Given: 𝐴𝐵 and 𝐴𝐶 are tangent to circle O at B and C.
Prove: 𝐴𝐵 ≅ 𝐴𝐶
PARTNER A
• Explain each step of your
reasoning, including the
justification for that step.
PARTNER B
• Re-state each step of your
partner’s reasoning, asking
clarifying questions.
Theorem 9-2
Theorem 9-2: If a line in the plane of a circle is
perpendicular to the radius at its outer endpoint, then
the line is tangent to the circle.
Given: Line 𝑙 is in the plane of ⊙ 𝐶
𝑙 ⊥radius 𝐴𝐶 at A
Prove: 𝑙 is tangent to ⊙ 𝐶
PARTNER B
• Describe the assumption
you will make.
PARTNER A
• Explain why your partner is
making this assumption.
• Re-state each step of your
partner’s reasoning, asking
clarifying questions.
• Explain each step of your
reasoning, including the
justification for that step.
• Explain clearly where the
contradiction is!
• What is the conclusion that
can be drawn?
Applying Theorems About Radii and Tangents
Explain and/or justify the steps you took to complete the following
problems:
Applying Theorems About Radii and Tangents
Explain and/or justify the steps you took to complete the following
problems:
Learning Log Summary
LT 3 - I can use indirect proof to show that
the tangent is perpendicular to the radius of
a circle drawn at the point of tangency.
To prove the tangent is perpendicular to the radius at
the point of tangency…
The contradiction in this proof is created by…
Learning Log Summary
LT 9 - I can construct a tangent through a
point on the circle and through a point
outside the circle.
LT 10 - I can explain the relevant geometry
involved in the construction of a tangent
line from a point outside of the circle.
Arcs and Central Angles – Basic Terms
A central angle of a circle is
an angle with its vertex at
the center of the circle.
An arc is an unbroken part
of the circle as a results of
drawing a central angle.
∠𝑩𝑶𝑪 is a central angle
Arcs and Central Angles – Basic Terms
Two points B and C on circle
O are always the endpoints of
two arcs. B and C and the
points on ⊙O in the interior of
∠𝐵𝑂𝐶 form a minor arc.
B and C and the remaining
points of ⊙O form a major
arc.
Arcs and Central Angles – Basic Terms
B
If B and C are the endpoints
of the diameter, then the
two arcs are called
semicircles.
C
Arcs and Central Angles – Basic Terms
The measure of a minor arc
is defined to be the
measure of its central angle.
The measure of a major arc
is 360° minus the measure of
the minor arc.
Arcs and Central Angles – Basic Terms
Adjacent arcs of a circle
are arcs that have exactly
one point in common.
Arcs and Central Angles – Basic Terms
Arc Addition Postulate:
The measure of the arc
formed by two adjacent
arcs is the sum of the
measures of these two arcs.
Arcs and Central Angles – Basic Terms
Congruent arcs are arcs, in
the same circle or in
congruent circles, that have
equal measures.
F
E
C
D
Caution: A congruent
central angle does not
guarantee congruent
arcs.
Theorem 9-3
Theorem 9-3: In the same circle or in congruent circles, two
minor arcs are congruent if and only if their central angles are
congruent.
List all pairs of congruent arcs in the diagram to the right and
explain how you know.
Theorem 9-4
In ⊙A shown to the right, 𝐷𝐵 cuts off two arcs,
𝐵𝐷 and 𝐷𝐹𝐵. We call 𝐵𝐷 the arc of chord 𝐷𝐵
Theorem 9-4
In the same circle or in congruent circles:
(1) Congruent arcs have congruent chords.
Partner A will prove this statement.
(2)Congruent chords have congruent arcs.
Partner B will prove this statement.
Theorem 9-5
A diameter that is perpendicular to a chord
bisects the chord and its arc.
Partners A/B will prove this statement together.
Theorem 9-6
In the same circle or in congruent
circles:
(1) Chords equally distant from the
center (or centers) are congruent.
Partner A will prove this statement.
(2)Congruent chords are equally
distant from the center (or centers)
Partner B will prove this statement
C
Learning Log Summary
LT 5 - I can apply theorems about the
chords of a circle in proof and problem
solving.
Theorems involving chords state…
The chord of an arc…
An inscribed angle is an
angle whose vertex is on a
circle and whose sides
contain chords of the circle.
The angle intercepts the arc
shown in in color.
Theorem 9-7
The measure of an inscribed angle is equal to half
the measure of its intercepted arc.
𝑚∠𝐵𝐶𝐷 =
1
𝑚𝐵𝐷
2
Theorem 9-7
The measure of an inscribed angle is equal to half the
measure of its intercepted arc.
Given: ⊙C
1
Prove: 𝑚∠𝐵𝐶𝐷 = 2 𝑚𝐵𝐷
Hint: Consider Triangles ACB and ACD.
How does angle 3 relate to angle 1?
How does angle 4 relate to angle 2?
Theorem 9-7
We can’t assume that C is in the interior of the inscribed
angle. Where else could it be?
C could lie on the
exterior of the inscribed
angle.
C could lie on one of the
sides of the inscribed
angle.
Theorem 9-7
Prove that the theorem holds true in each of these cases.
C lies on the exterior of
the inscribed angle.
C lies on one of the sides
of the inscribed angle.
Theorem 9-7
Explain, in a paragraph, why each of the corollaries to
Theorem 9-7 must be true.
• If two inscribed angles intersect the same
arc, then the angles are congruent.
• An angle inscribed in a semicircle is a right
angle.
• If a quadrilateral is inscribed in a circle,
then its opposite angles are
supplementary.
Theorem 9-8
The measure of an angle formed by a chord and a tangent
is equal to half the measure of the intercepted arc.
If 𝐴𝐵 is a chord and 𝐴𝐶 is tangent
1
to ⊙, then 𝑚∠𝐵𝐴𝐶 = 2 𝑚𝐵𝐶
Sketchpad Demo
Learning Log Summary
LT 4 - I can prove and apply relationships
between arcs, central, inscribed [interior and
exterior] angles.
An angle inscribed in a circle…
The arc intercepted by a central angle / inscribed
angle…
Quiz Overview
Learning Targets:
1- Tangents and Radii
2/3 – Proof by Contradiction
4 – Arcs/Central and Inscribed Angles
5 – Chords of a Circle
6 – Inscribed Polygons
9/10 – Tangent Constructions
Theorems:
9-1 through 9-8
Problem Solving
Circle D has center (-2, -7) and radius 7.
What is the measure, in degrees, of the
major arc that passes through points
H(-2, 0), J(5, -7), and K(-9, -7)?
Problem Solving
Find ????.
Problem Solving
𝐵𝐶 is tangent to ⊙A at B and
⊙D at C. If AB = 12, BC = 18,
and DC = 3, what is the exact
length of AD?
Problem Solving
In this diagram, 𝐴𝐵 = 30, 𝑂𝑀 = 20,
and 𝑂𝑁 = 18. What is 𝐶𝑁?
Problem Solving
In this diagram, BE is a radius and is
perpendicular to chord AC. 𝐴𝐵 = 10
and 𝐴𝐶 = 16. Find DE.
Problem Solving
Find 𝑚∠𝐹.
Problem Solving
Find 𝑚𝐴𝐵.