Geometry - Loudoun County Public Schools



NAME _________________________________ BLOCK __________

Chapter 10

Properties of

Circles

Sections Covered:

10.1 Use Properties of Tangents

10.2 Find Arc Measures

10.3 Apply Properties of Chords

10.4 Use Inscribed Angles and Polygons

10.5 Apply Other Angle Relationships in Circles

10.7 Write and Graph Equations of Circles

10.6 Find Segment Lengths in Circles

|◄ |~ April 2015 ~ |► |

|Sun |Mon |Tue |Wed |

|RADIUS | | | |

|CHORD | | | |

|DIAMETER | | | |

|SECANT | | | |

|TANGENT | | | |

|POINT OF TANGENCY | | | |

Discovery: Draw the following pairs of circles. Determine if the tangents are internal or external tangents.

1. non-intersecting circles with 4 common tangents. 2. non-intersecting circles with no common tangents.

3. 1 point of intersection with 1 common tangent. 4. 1 point of intersection with 3 common tangents.

5. intersecting circles with 2 common tangents.

Theorem: In a plane, a line is _________________ if and only if the line is

_________________________ to a radius of a circle at its endpoint on the circle.

Is line m tangent to ⊙Q? Why or why not?

Practice with Tangents:

Ex1: In the diagram, B is a point of tangency. Ex2: [pic] is a tangent to ⊙C. Find x.

Find the radius r of ⊙C.

Ex3: [pic]is a radius of ⊙C. Determine whether

[pic] is a tangent to ⊙C. Explain your reasoning.

Theorem: Tangent segments from a common external point are _________________.

If [pic]and [pic]are tangent segments, then ______________.

Practice with Two Tangents:

Ex4: [pic] is tangent to ⊙C at S and [pic]is Ex5: Find the perimeter of the polygon.

tangent to ⊙C at T. Find the value of x.

Standard Equation of a Circle:

Circles which are centered at the origin (0, 0) have simple equations. (h, k) = (0, 0)

[pic]

Example 1

Example 2

Example 3

Recall: Distance Formula

.

Key Concept: There are different types of angles that can lie on or within

circle. Additionally, any part of the circle is called an ________________.

New Vocabulary for the Parts of a Circle:

|NAME |DEFINITION |NAMING |EXAMPLE BASED ON THE |

| | | |PICTURE ABOVE |

|CENTRAL ANGLE | | | |

|MINOR ARC | | | |

|MAJOR ARC | | | |

|SEMICIRCLE | | | |

Note: Based on the above picture, we say that [pic]is the _________________________ of[pic].

Ex1: Find the measure of each arc of ⊙P. where [pic]is a diameter.

a. m[pic] b. m [pic] c. m [pic]

Ex2: Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc.

a. m [pic] b. m [pic] c. m [pic]

d. m [pic] e. m [pic] f. m [pic]

On Your Own: In the figure, [pic]and [pic]are diameters of ⊙U. Find the measures of the indicated arcs. (Hint: begin by filling in all the missing angles based on the directions you were just given. Please put a star at the top of these notes if you read all these directions.)

1. m[pic]= _________ 2. m[pic]= _________ 3. m[pic]= _________

4. m[pic]= _________ 5. m[pic]= _________ 6. m[pic]= _________

7. m[pic]= _________ 8. m[pic]= _________ 9. m[pic]= _________

10. m[pic]= _________

Congruent Circles:

Congruent Arcs:

Ex3: Tell whether[pic]. Explain.

1. 2.

3. 4.

Arc of the chord-

Ex6: Ex7:

Ex8: Ex9:

a. [pic]

b. [pic]

c. [pic]

Ex10:

Inscribed Angle:

Intercepted Arc:

Ex1: Identify Inscribed Angles: Circle every inscribed angle below.

a. b. c. d.

Ex2: Practice Inscribed Angles: Find the following measures. You must always write the formula/equation.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

On Your Own: Find the indicated measure in ⊙M.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

Ex4: Find [pic] and [pic].

1. 2.

Practice: Quadrilaterals Inscribed in a Circle:

Ex5: Find the value of each variable.

1. 2. 3.

|Picture: | | | |

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|Work: | | | |

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|Line Types | | | |

|Vertex Location: | | | |

|Angle/Arc Relationship: | | | |

|Picture: | | | |

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|Work: | | | |

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|Line Types | | | |

|Vertex Location: | | | |

|Angle/Arc Relationship: | | | |

Practice with Angle Relationships in Circles: Identify the location of the vertex as CENTER, ON, IN, or OUT. Then, find the missing angles and arcs. You MUST write the formula and then solve the equation.

1. Location: 2. Location: 3. Location:

_________ _________ _________

Formula: Formula: Formula:

4. Location: 5. Location:

_________ _________

Formula: Formula:

6. Location: 7. Location:

_________ _________

Formula: Formula:

8. Location: 9. Location:

_________ _________

Formula: Formula:

[pic]

| | |

|What do you see? |What do you see? |

|What’s the formula? |What’s the formula? |

| |

|What do you see? |

|What’s the formula? |

Ex1: Practice with Segments in Circles: Identify the type of segments as 2 CHORDS, 2 SECANTS, OR SECANT AND TANGENT. Then, find the variable. You MUST write the formula and then solve the equation.

1. Type: 2. Type:

Formula: Formula:

3. Type: 4. Type:

Formula: Formula:

On Your Own: Identify the type of segments as 2 CHORDS, 2 SECANTS, OR SECANT AND TANGENT. Then, find the variable. You MUST write the formula and then solve the equation.

1. Type: 2. Type: 3. Type:

Formula: Formula: Formula:

4. Type: 5. Type:

Formula: Formula:

6. Type: 7. Type:

Formula: Formula:

-----------------------

[pic]

Internal -

External -

Key Concept: If [pic]is a tangent, then [pic] and [pic]are ___________. So, this is a ______________. So, the side lengths must fit __________________.

A

B

C

8

6

x

A

B

C

[pic]

[pic]

[pic]

Theorem: In the same circle, or in congruent circles, two minor arcs are ________ if and only if their corresponding chords are ________.

Statement:

Theorem: If one chord is a ____________________________ of another chord, then the first chord is a __________________.

Statement:

Theorem: If a diameter of a circle is ____________ to a chord, then it ____________ the chord and its arc.

Statement:

Theorem: In the same circle or in two congruent circles, two chords are ________________if and only if they are __________________ from the center.

Statement:

Theorem: The measure of an inscribed angle is ____________angle is _____________ of its intercepted arc.

Formula which must always be written and substituted in for:

Statement:

∙

∙

∙

∙

Theorem: A triangle inscribed in a semicircle is ALWAYS a ________________ triangle where the _________________is ALWAYS the ____________________.

Theorem: If two __________________ angles of a circle intercept the same arc, then the angles are ___________________.

Statement:

Theorem: A quadrilateral can be inscribed in a circle if and only if its opposite angles are ____________________.

Statement:

Common Mistake:

Not writing the equation.

**Extremely important to substitute correctly into formula

*Then solve the equation.

*Remember to put arc in for the word arc & angle in for the word angle.

Common Mistake: Some people will substitute in for “TOTAL” incorrectly.

They will write the total of 12 and x as 12x. This is the product. Remember that TOTAL means addition so it should be 12 + x.

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