Student Name

Hour ______

Chapter 10 – Circles

Geometry

2

10-1 Circles and Circumference I can…

Identify and

use parts of a

circle.

Solve

problems

involving the

circumference

of a circle.

Circle – a set of points equidistant from a single point.

Parts of a Circle – label diagram with

each part.

Center Of A Circle

Chord

Radius

Diameter

Secant

Tangent

Example 10-1-1: Identify Segments in a Circle

A. Name the circle, and identify a radius and a

chord.

Chord Radius

B. Identify a chord, a radius and a diameter of the circle.

Chord Radius Diameter

G

3

Example 10-1-2: Find Radius and Diameter

If RT = 21 cm, what is the length of 𝑄𝑉̅̅ ̅̅ ?

Example 10-1-3: Find Measures in Intersecting Circles

The diameter of X is 22 units, the diameter

of Y is 16 units, and WZ = 5 units. Find XY.

Example 10-1-4: Real World

The Unisphere is a giant steel globe that sits in Flushing Meadows-Corona Park

in Queens, New York. It has a diameter of 120 feet. Find its circumference.

4

Example 10-1-5: Find Diameter and Radius

Find the diameter and the radius of a circle to the nearest hundredth if the

circumference of the circle is 65.4 feet.

𝑑 = ____________ 𝑟 =____________

Example 10-1-6: Circumference of Circumscribed Polygon

Find the exact circumference of K.

Summarize the Lesson:

5

10-2 Measuring Angles and Arcs I can…

Identify

central

angles, major

arcs, minor

arcs, and

semicircles,

and find their

measures.

Find arc

lengths.

Example 10-2-1: Find Measures of Central Angles

Find the value of x.

Example 10-2-2: Classify Arcs and Find Arc Measures

A. 𝑊𝐶̅̅ ̅̅ ̅ is a radius of C. Identify 𝑋𝑍�̂� as a major

arc, minor arc, or semicircle. Then find its

measure.

B. 𝑊𝐶̅̅ ̅̅ ̅ is a radius of C. Identify 𝑊𝑍�̂� as a major

arc, minor arc, or semicircle. Then find its

measure.

6

Example 10-2-2: Classify Arcs and Find Arc Measures

C. 𝑊𝐶̅̅ ̅̅ ̅ is a radius of C. Identify 𝑋�̂� as a major

arc, minor arc, or semicircle. Then find its

measure.

Example 10-2-3: Find Arc Measures in Circle Graphs

A. Find the measure of 𝑚𝐾�̂�.

B. Find the measure of 𝑚𝑁𝐽�̂�.

Example 10-2-4: Use Arc Addition to Find Measures of Arcs

A. Find 𝑚𝐿𝐻𝐼̂ in M.

B. Find 𝑚𝐼𝐽�̂� in M.

7

Example 10-2-5: Find Arc Length

A. Find the length of 𝐷�̂�. Round to the nearest hundredth.

B. Find the length of 𝐷�̂�. Round to the nearest hundredth.

C. Find the length of 𝐷�̂�. Round to the nearest hundredth.

8

10-3 Arcs and Chords I can…

Recognize and

use

relationships

between arcs

and chords.

Recognize and

use

relationships

between arcs,

chords, and

diameters.

Example 10-3-1: Use Congruent Chords to Find Arc Measure

A circular piece of jade is hung from a chain by two wires around the stone.

𝐽𝑀̅̅ ̅̅ ≅ 𝐾𝐿̅̅ ̅̅ and 𝑚𝐾�̂� = 90. Find 𝑚𝐽�̂�.

Example 10-3-2: Use Congruent Arcs to Find Chord Lengths

In the figure, 𝐴 ≅ 𝐵 and 𝑊�̂� ≅ 𝑌�̂�. Find 𝑊𝑋.

9

Example 10-3-3: Use a Radius Perpendicular to a Chord

In G, 𝑚𝐷𝐸�̂� = 150. Find 𝑚𝐷�̂�.

Example 10-3-4: Use a Diameter Perpendicular to a Chord

In the ceramic stepping stone below, diameter 𝐴𝐵̅̅ ̅̅ is 18

inches long and chord 𝐸𝐹̅̅ ̅̅ is 8 inches long. Find CD.

Example 10-3-5: Chords Equidistant from Center

In P, 𝐸𝐹 = 𝐺𝐻 = 24. Find 𝑃𝑄.

Summarize the lesson:

10

10-4 Inscribed Angles I can…

Find measures

of inscribed

angles.

Find measures

of angles of

inscribed

polygons.

Example 10-4-1: Use Inscribed Angles to Find Measures

A. Find 𝑚∠𝑋.

B. Find 𝑚𝑌�̂�.

Example 10-4-2: Use Inscribed Angles to Find Measures

Find 𝑚∠𝑅.

11

Example 10-4-3: Use Inscribed Angles in Proofs

Write a two-column proof. Given: 𝐿�̂� ≅ 𝑀�̂�

Prove: 𝛥𝑀𝑁𝑃 𝛥𝐿𝑂𝑃

Statements Reasons

1. 𝐿�̂� ≅ 𝑀�̂� 1.

2. 2. If minor arcs are congruent, then

corresponding chords are congruent.

3. ∠𝑀 intercepts 𝑁�̂� and

∠𝐿 intercepts 𝑁�̂�. 3.

4. ∠𝑀 ≅ ∠𝐿 4.

5. ∠ ≅ ∠ 5.

6. 6. AAS ≅Theorem

Example 10-4-4: Find Angle Measures in Inscribed Triangles

Find 𝑚∠𝐵.

12

Example 10-4-5: Find Angle Measures

An insignia is an emblem that signifies rank, achievement,

membership, and so on. The insignia shown is a quadrilateral

inscribed in a circle. Find mS and mT.

Example 10-4-6: More Algebra Practice

A. Find x. B. Find x and y.

C. Find x and y.

3x C

A

B T S

U R

x

y

80 85

D A

C B 40x 19y

10y

22x P

13

10-5 Tangents

I can…

Use properties

of tangents.

Solve

problems

involving

circumscribed

polygons.

Point of Tangency – the point at which a _____________ __________

intersects the ________ to which it is __________.

Tangent Circles

Internally tangent circles Externally tangent circles

Concentric Circles are coplanar circles with the same center.

Common Tangent

Common internal tangent Common external tangent

Example 10-5-1: Identify Common Tangents

Draw the common tangents. If no common tangent exists, state no common

tangent.

A. B. C.

14

Example 10-5-2: Identify a Tangent

𝐾𝐿̅̅ ̅̅ is a radius of K. Determine whether 𝐿𝑀̅̅ ̅̅ is tangent

to K. Justify your answer.

Example 10-5-3: Use a Tangent to Find Missing Values

In the figure, 𝑊𝐸̅̅ ̅̅ ̅ is tangent to D at W. Find 𝑥.

Example 10-5-4: Use Congruent Tangents to Find Measures

𝐴𝐶̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ are tangent to Z. Find 𝑥.

15

Example 10-5-5: Find Measures in Circumscribed Polygons

The round cookies are marketed in a triangular

package to pique the consumer’s interest. If

∆𝑄𝑅𝑆 is circumscribed about T, find the

perimeter of ∆𝑄𝑅𝑆.

Example 10-5-6: Tangent

Is 𝐶𝐸̅̅̅̅ tangent to D? Explain.

Summarize the lesson:

E C

D

11

43

45

16

10-6 Secants, Tangents, & Angle Measures I can…

Find measures

of angles

formed by lines

intersecting on

or inside a

circle.

Find measures

of angles

formed by lines

intersecting

outside the

circle.

Example 10-6-1: Use Intersecting Chords or Secants

A. Find x.

B. Find x.

C. Find x. D. Your turn. Find x.

17

Example 10-6-2: Use Intersecting Secants and Tangents

A. Find 𝑚∠𝑄𝑃𝑆.

B. Find 𝑚𝐵𝐶�̂�.

Example 10-6-3: Use Tangents and Secants that Intersect Outside a Circle

A. Find 𝑚𝐵�̂�.

B. Find 𝑚𝑋𝑌�̂�.

18

Example 10-6-4: Apply Properties of Intersecting Secants

The diagram shows the path of a light ray as it hits a cut

diamond. The ray is bent, or refracted, at points A, B, and C. if

𝑚𝐴�̂� = 96° and 𝑚∠𝑆 = 35°, what is 𝑚𝑅𝐵�̂�?

Example 10-6-5: Apply Properties of Intersecting Secants

19

10-7 Special Segments in Circles I can…

Find measures

of segments

that intersect

in the interior

of a circle.

Find measures

of segments

that intersect

in the exterior

of a circle.

Example 10-7-1: Use the Intersection of Two Chords

A. Find 𝑥.

B. Find 𝑥.

Example 10-7-2: Find Measures of Segments in Circles

Biologists often examine organisms under microscopes.

The circle represents the field of view under the

microscope with a diameter of 2 mm. Determine the

length of the organism if it is located 0.25 mm from the

bottom of the field of view. Round to the nearest

hundredth.

20

Example 10-7-3: Use the Intersection of Two Secants

Find 𝑥.

Example 10-7-4: Use the Intersection of Two Secants

Find 𝑥.

Example 10-7-5: Use the Intersection of a Secant and a Tangent

𝐿𝑀̅̅ ̅̅ is tangent to the circle. Find 𝑥. Round to the

nearest tenth.

D H

F

G

E

10

11

12 x

21

10-8 Equations of Circles

I can…

Write the

equation of a

circle.

Graph a

circle on the

coordinate

plane.

Example 10-8-1: Write an Equation Using the Center and Radius

A. Write the equation of the

circle with a center at

(3, – 3) and a radius of 6.

B. Write the equation of the circle

graphed.

Example 10-8-2: Write an Equation Using the Center and a Point

Write the equation of the circle that has its center at (– 3, – 2) and passes through

(1, – 2).

Example 10-8-3: Graph a Circle – May skip?

The equation of a circle is x2 – 4x + y2 + 6y = –9. State the coordinates of the center

and the measure of the radius. Then graph the equation.

22

Example 10-8-5: Intersections with Circles

Find the point(s) of intersection between 𝑥2 + 𝑦2 = 32 and 𝑦 = 𝑥 + 8.

Graphically – this is the answer.

Algebraically: Use substitution.