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Unit 11Segments and Equations of Circles

Lesson 1: Properties of Tangents

Opening Exercise

Draw 3 different diagrams of a circle and a line given the following:

They do NOT intersect. They intersect once. They intersect twice.

A line that intersects a circle at exactly two points is called a ______________________________ line.

A line that intersects a circle at exactly one point is called a _______________________________ line.

1

Example 1

You will need a protractor.

In the accompanying diagram, P is called:

Using a protractor, measure the angle formed by the radius and the tangent line. Write the angle measure on the diagram.

Will this work for all angles formed by a radius and a tangent line?

Important Discovery!

A tangent line to a circle is ________________________________ to the radius of the circle drawn to the point of tangency.

The converse is also true. So, a line through a point on a circle is tangent at the point if, and only if, it is perpendicular to the radius drawn to the point of tangency.

4 Common Tangents(2 completely separate circles)

3 Common Tangents(2 externally tangent circles)

2 Common Tangents(2 overlapping circles)

1 Common Tangent(2 internally tangent circles)

0 Common Tangents

(2 concentric circles)Concentric circles are circles

with the same center.

(one circle floating inside the other,

without touching)

Tangent lines that meet two circles are called _____________________________ tangents. Listed below are the 5 different ways we can discuss common tangents.

2

Example 2

In the diagram, and are tangent to circle A at points D and E respectively.

Write a two-column proof to prove .

Statements Reasons

1. and are tangent to circle A at 1. Given points D and E respectively

2. and are right angles 2.

3. and are right triangles. 3.

4. 4.

5. 5.

6. 6.

7. 7.

3

Important Discovery!

Theorem

The two tangent segments to a circle from an exterior point are ________________________.

CD CE

CD CE

CD CE

ADC AEC

ADC AEC

AD AE

AC AC

ADC AEC

CD CE

Example 3

In circle A, the radius is and .

a. Find .

b. Find . Explain how you know.

c. Find . Explain how you know.

d. Find the area of .

e. Find the perimeter of quadrilateral .

Example 4

If , , and , is tangent to circle A at point B? Explain.

4

9 mm 12 mmBC

AC

AD

CD

ACD

ABCD

5AB 12BC 13AC

Example 5

You will need a compass and a ruler.

Construct a line tangent to circle A through point B.

5

Exercises

1. In the diagram, circle O is inscribed in so that the

circle is tangent to  at F, to at E, and to at D.  If and , find the perimeter of .

2. In circle A, , , and .

a. Find the length of the radius of the circle.

b. Find (to the nearest tenth).

c. Find

6

ABC

ABC

12EF 13AE : 1:3AE AC

BC

EC

Homework

1. If , , and , is tangent to circle A? Explain.

2. In the given figure, the three segments are tangent to the circle at point F, B and G. Find DE.

3. In the given figure, circles X and Y have two tangents drawn to them from external point T. The points of tangency are C, A, S, and E. The ratio of TA to AC is 1:3. If TS =24, find the length of SE.

7

9BC 6AB 15AC

Lesson 2: Tangent Segments and Angles

Opening Exercise

Find x if the line shown is tangent to the circle at point B.

8

Example 1

Given circle A with tangent .

a. Draw . What is the measure of ? Explain.

b. What is the measure of ? Explain.

c. Express the measure of the remaining two angles of in terms of a. Explain.

d. What is the measure of in terms of a? Show how you calculated your answer.

e. Summarize what we have just proven.

9

ABC BAC

ABG

ABC

BAC

Exercises

1. Solve for a. 2. Solve for a.

3. Solve for a.

10

Theorem

An inscribed angle formed by a secant and a tangent line is ______________________ of the angle measure of the arc it intercepts.

We have learned a lot about tangents! Here is a summary:

A tangent line intersects a circle at exactly one point (and is in the same plane). The point where the tangent line intersects a circle is called the point of tangency. The tangent line is perpendicular to a radius whose endpoint is the point of

tangency. The two tangent segments to a circle from an exterior point are congruent. The measure of an angle formed by a tangent segment and a chord is one-half the

angle measure of its intercepted arc. If an inscribed angle intercepts the same arc as an angle formed by a tangent

segment and a chord, then the two angles are congruent.

Example 2

Find the values of a, b, and c.

11

Example 3

Find the values of a, b, and c.

12

Homework

1. Calculate the value of z. 2. Find the values of a and b.

3. Complete the following two-column proof.

Given: Circle P with tangents AC and AB Ray AP is drawn

Prove: bisects

Statements Reasons

1. Circle P with tangents AC and AB 1. GivenRay AP is drawn

2. Draw radii BP and CP 2. Auxiliary Lines

3.

13

AP CAB

Lesson 3: Interior and Exterior Angles

Opening Exercise

Vocabulary

Definition Diagram

Secant Line

a line that intersects a circle in exactly two points

What is the difference between a tangent and a secant?

On the given circle, draw two secants that:

a. intersect inside the circle. b. intersect outside the circle.

c. intersect on the circle. d. do not intersect.

14

Example 1

Using our knowledge of inscribed angles, we are going to find the measure of an interior angle that is not a central angle.

To find x, draw chord BD.

Can you determine any of the angle measures in ? Explain.

Find x. Justify your answer.

Interior Angle (vertex inside the circle)

Formula

15

Example 2

a. Find the value of x: b. Find the value of x:

Exercises

1. Find the values of angles x and y. 2. Find the value of x.

16

Exterior Angle (vertex outside the circle)

Formula

Example 3

Write the equation used to find in the following diagrams:

17

Exercises

1. Find the measure of ∠BCE .

2. Find the measure of .

18

Homework

1. Find the value of x. 2. Find the measure of ∠DEB .

3. Find the measure of ∠E

.

19

Lesson 4: Interior and Exterior Angles II

Opening Exercise

Find the value of x in the diagrams pictured below:

20

Example 1

Find the value of x.

Example 2

If , solve for x.

21

28m DCE

Example 3

Find the values of x and y.

22

Exercises

In the following questions, find the value of x:

1.

2.

23

Homework

1. Find the values of x and y.

2. Find the value of x.

3. Find .

24

Lesson 5: Similar Triangles in Circle–Secant Diagrams

Opening Exercise

Given: Circle with chords and

intersecting at point F

Prove:

25

BD CE

Exercises

1. Find the value of x.

2. In the circle shown, , ,

and . Find the shorter part of .

26

Intersecting Chords

Formulad

c b

a

11DE 10BC

8DF

Segment Lengths

It is also true that when secant lines, tangent lines, or secant and tangent lines intersect outside of a circle, their segment lengths can be

found using: .

In words, .

Sometimes the whole and the outside piece are one in the same. In this case, the formula is

.

In words, .

27

a a b c c d

2a b b c

Exercises

1. Find the value of x in simplest radical form.

2. If , , and , find .

3. Find the value of x.

28

6CE 9CB 18CD CF

Homework

1. Find the value of x.

2. Find the value of x.

3. Find the value of x.

29

2x

3

6

5

Lesson 6: Writing the Equation of a Circle

Opening Exercise

a. Find the length of the line segment shown on the coordinate plane below.

b. Using the distance formula, find the distance between the points and .

30

9,15 3,7

Example 1

If we graph all of the points whose distance from the origin is equal to 5, what shape will be formed?

Using the given coordinate plane, plot 4 points that are 5 units away from the origin.

Now, we need to find 4 more.

Write down any ideas that you might have to find the location of the next point that is also 5 units from the origin.

Compare your plan with a partner. Once you agree on a plan, plot three more points using this method. Using your compass, connect these points to form a circle.

In the above circle, the center is located at ___________________ and the radius length is ________.

We found the location of a point on the circle by using

_______________________________________________, which states ___________________________________.

If we generalize this formula by using a point named , the point will satisfy the

equation when the circle has a center at the origin.

31

,x y

2 2 25x y

Example 2

Now, let’s look at a circle that is not centered at the origin.

This circle is centered at and has a radius length of 5 units.

Is this circle congruent to the circle we constructed?

Is there a sequence of basic rigid motions that would take this circle center to the origin? Explain.

The equation for this circle can be found using this same pattern to move the center of the circle back to the origin.

The equation of this circle is:

32

Standard Form of the Equation of a Circle

with center and radius length

2,3

2 2 2x a y b r ,a b r

Example 3

Write the equation of the circle that is graphed below.

Example 4

Find the radius and center of the circle given by the equation: 2 212 4 81x y

Example 5

Write an equation for the circle whose center is at 9,0 and has radius 7.

33

Homework

1. Describe the circle given by the equation: 2 27 8 9x y .

2. Write the equation for a circle with center 0, 4 and radius 8.

3. Write the equation for the circle shown.

34

Lesson 7: Writing the Equation of a Circle II

Opening Exercise

Two points in the plane, 3,8A and 17,8B

, represent the endpoints of the diameter of a circle.

a. What is the center of the circle? Explain.

b. What is the radius of the circle? Explain.

c. Write the equation of the circle.

35

Example 1

Write the equation of a circle with center that passes through ?

Example 2

A circle with center is tangent to the x-axis.

a. What is the radius of the circle?

b. What is the equation of the circle?

36

3,10 12,12

2, 5

Example 3

Given a circle centered at the origin that goes through point (0, 2), determine whether or not

this circle would go through the point .

Example 4

Determine the center and radius of each circle:

a.

b.

c.

37

2 24 6 50x y

2 23 3 75x y

2 24 2 4 9 64 0x y

38

Homework

1. Determine the center and radius of the circle .

2. Write the equation of a circle that has a center of (-4, -3) and is tangent to the y-axis.

3. A circle has a diameter with endpoints at and . Write the equation for this circle.

39

Lesson 8: Recognizing Equations of Circles

Opening Exercise

Complete the following table:

Polynomial Factored Form

40

Example 1

Find the center and the radius of the following:

a.

b.

Example 2

Find the center and the radius of the following:

Example 3

41

Equation of a Circle

Standard Form General Form

2 24 4 6 9 36x x y y

2 210 25 14 49 4x x y y

2 24 12 41x x y y

Could the circle with the equation have a radius of 4? Why or why not?

Example 4

Identify the graphs of the following equations as a circle, point, or an empty set.

a. b.

Summary

When is … The figure is …

Positive

Negative

Zero

42

2 26 7 0x x y

2 2 4 0x y x 2 2 6 4 15 0x y x y

Exercises

1. The graph of the equation below is a circle. Identify the center and radius of the circle.

2. Identify the graphs of the following equations as a circle, point, or an empty set.

a.

b.

c.

Example 5

Chante claims that two circles given by and are externally tangent. She is right. Show that she is.

43

2 210 8 8 0x x y y

2 22 1x x y

2 2 3x y

2 2 6 6 7x y x y

2 22 4 49x y

Homework

1. Identify the center and radius of the following circles.

a.

b.

c.

d.

2. Sketch a graph of the equation .

44

2 225 1x y

2 22 8 8x x y y

2 220 10 25 0x x y y

2 2 19x y

2 2 14 16 104 0x y x y

Lesson 9: Inscribed and Circumscribed Circles

Opening Exercise

In each diagram, try to draw a circle with center D that is tangent to both rays of .

Which diagrams did it seem impossible to draw such a circle? Why did it seem impossible?

What do you conjecture about circles tangent to both rays of an angle? Why do you think that?

45

Important Discovery!

If a circle is tangent to both rays of an angle, then the center of the circle lies on the:

Example 1

You will need a compass and a straightedge.

Construct a circle that is tangent to both rays of the given angle.

1. How do you find the center?

2. How do you find the radius?

Now let’s make the construction!

46

Example 2

You will need a compass and a straightedge.

Let’s construct a circle inscribed in a triangle!

In the space below, using a straightedge, draw a large triangle.

a. Pick any two angles and construct their angle bisectors.

b. What is special about the intersection point of these angle bisectors?

c. Construct a perpendicular segment from this intersection point to any side of your triangle. What is this segment called?

d. Using your compass, the intersection point of your angle bisectors, and this segment length, construct a circle. This is called the incircle.

47

Theorems

If a circle is tangent to both rays of an angle, then its center lies on the angle bisector.

Every triangle contains an inscribed circle whose center is the intersection of the triangle’s angle bisectors.

We have now discussed points of concurrency in triangles over the course of the year. Let’s take a look at them one more time to see how this relates to inscribed and circumscribed circles.

Draw in the points of concurrency in the diagrams below:

Centroid Incenter Circumcenter Orthocenter

medians angle bisectorsperpendicular

bisectorsaltitudes

Example 3

You will need a compass and a straightedge.

Construct a circle so that it is circumscribed around the triangle pictured. This is called the circumcircle.

Recall: To find the inscribed circle, we used incenter.

To find the circumscribed circle, we will use __________________________________ .

48

Exercises

1. Point B is the centroid. Find x, y, and z.

2. Point A is the circumcenter. Find x, y and z.

49

Homework

1. Draw the incircle to the pictured triangle:

2. Draw the circumcircle to the pictured triangle:

50

Lesson 10: Cyclic Quadrilaterals

Opening Exercise

The above 4 diagrams are examples of cyclic quadrilaterals. What do you think the definition of a cyclic quadrilateral is?

What is another term that we have previously used for diagrams like the cyclic ones above?

51

Example 1

Given cyclic quadrilateral ABCD shown in

the diagram, prove that .

52

If a quadrilateral is cyclic, then its __________________________________ angles are

__________________________________________.

Exercises

1. What is the exact value of x that guarantees that the quadrilateral shown in the diagram is cyclic?

2. Quadrilateral BDCE is cyclic, O is the center of the circle, and . Find .

3. In the diagram, and .

a. Find the values of s and t.

b. What kind of figure is the quadrilateral BCDE? How do you know?

53

130m BOC m BEC

72m BED

Homework

1. In the diagram given, is the diameter,

, and . Find .

2. In circle A, . Find .

3. In the diagram given, quadrilateral JKLM is cyclic. Find the value of n.

54

BC

25m BCD CE DE m CED

15m ABD m BCD