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Geometry

Unit 5B Circles

Learning Target #1:

Tangent and Chord Properties

Learning Target #2:

Tangent and Chord Inside and Outside Segments

Learning Target #3:

Arc Length

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Geometry Unit 5B: Circles Segments Notes

Tangent Properties

Name Theorem Hypothesis Conclusion

Perpendicular Tangent

Theorem

If a line is tangent to a

circle, then it is

perpendicular to the

radius drawn to the point

of tangency.

Converse of

Perpendicular Tangent

Theorem

If a line is perpendicular

to a radius of a circle at

a point on the circle,

then the line is tangent to

the circle.

Example: Is AB tangent to Circle C? Example: Find ST.

Name Theorem Hypothesis Conclusion

Tangent Segments

Theorem

If two segments are

tangent to a circle from

the same external point,

then the segments are

congruent.

Example: Find perimeter of triangle. Example: Find DF if you know that DF and DE are tangent to C .

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Tangent Properties Practice

For problems 1-2, find the perimeter of each polygon.

1. 2.

3. Find the missing segment length. 4. JG is the diameter of the circle whose radius

is 11. If PG = 20 and JP = 30, is GP tangent to the circle?

5. Solve for x. 6. Given CD = 3(2x-2) and CB = -5x+16, find mCD.

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7. Is HM tangent to the circle? 8. Is FG tangent to the circle?

9. Find the value of x: 10. Is AB tangent to the circle?

11.Find the length of ? 12. What is the perimeter?

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Skills Practice

1. In the diagram below, AB = BD = 5 and AD = 7. Is BD tangent to C ? Explain.

2. AB is tangent to C at A and DB is tangent to C at D. Find the value of x.

a. b.

3. AB and AD are tangent to .C Find the value of x.

a. b.

4. AB is tangent to .C Find the value of r.

a. b.

5. Tell whether AB is tangent to .C Explain your reasoning.

a. b.

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Chord Properties

Name Theorem Hypothesis Conclusion

Congruent Angle-

Congruent Chord

Theorem

Congruent central

angles have congruent

chords.

Congruent Chord-

Congruent Arc Theorem

If two chords are congruent in

the same circle or two

congruent circles, then the

corresponding minor arcs are

congruent.

Example: Find the measure of arc HY and HYW. Example: Find the measure of arc YZ if the

measure of arc XW = 95

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Name Theorem Hypothesis Conclusion

Diameter-Chord

Theorem

If a radius or diameter is

perpendicular to a

chord, then it bisects the

chord and its arc.

Converse of Diameter-

Chord Theorem

If a segment is the

perpendicular bisector of

a chord, then it is the

radius or diameter.

Example: Find the measure of SQ. Example: Find the measures of arc PM, NP, and NM.

Example: Find the measure of HT. Then find the Example: Find the measures of arc CB, BE, and CE.

measure of WA if you know XZ = 6.

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Name Theorem Hypothesis Conclusion

Radius-Chord Theorem

If a radius (or part of a

radius) is Perpendicular

to a chord, then it

bisects the chord

Converse of Diameter-

Chord Theorem

If a radius (or part of a

radius) bisects a chord,

then it is Perpendicular

to the chord

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Name Theorem Hypothesis Conclusion

Equidistant Chord

Theorem

If two chords are

congruent, then they

are equidistant from the

center.

Converse of Equidistant

Chord Theorem

If two chords are

equidistant from the

center, then the chords

are congruent.

Example: Find EF. Example: Solve for x AND find the measure .AB

Find the measure of YX. Example: Solve for x.

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Practice - Chord Properties

Find the value of x in each circle. When necessary, find the measure of the indicated segment.

1. x = ________ 2. x = _________ 3. x = ________ AB = _______

4. Find QS. 5. Find mMN . 6. Solve for x.

7. In ,P QR = 7x – 20 and TS = 3x. What is x? 8. In ,K JL LM , KN = 3x – 2, and KP = 2x + 1.

What is x?

9. In ,O MO = 6 and LN = 16. Find x. 10. Solve for x.

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Segment Lengths (Inside of a Circle) NOTES

Name Theorem Hypothesis Conclusion

Segment Chord

Theorem

If two chords in a circle

interest, then the product of

the lengths of the segments of

one chord is equal to the

product of the lengths of the

segments of the second

chord.

Example: Find x. Example: Find x.

Example: Find x. Example: Find x.

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Skills Practice – Chord (Inside Segments)

1. Solve for x. 2. Solve for x.

3. Solve for x. 4 Solve for x.

5. Solve for x. 6. Solve for x.

7. Solve for x. 8. Solve for x.

x

6

15 35

x

8

24 36

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18

6

x

8

3

4

x

Segment Lengths (Outside Circle) NOTES

Secant Segment

Theorem

If two secant segments

intersect in the exterior of a

circle, then the product of the

lengths of the secant segment

and its external secant

segment is equal to the

product of the lengths of the

second secant segment and

its external secant segment.

Example: Find x. Example: Find x.

Example: Find x Example: Find x.

Example: Find x Example: Find x.

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Skills Practice – Chord (Inside Segments)

1. Solve for x 2 Solve for x

3. Solve for x 4. Solve for x

5. Solve for x 6. Solve for x

7. Solve for x 8. Solve for x

x

8

6

x

27

9

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Segment in Circles WITH Quadratics NOTES

Review – Combine Like Term

Directions: Distribute and collect like terms.

a. 8(21) = (x + 1 )(x + 18) b. 4(10) = x(x + 3)

Try these

1. (3x – 3)(3x – 3) = 4(9) 2. 5(8) = (2x – 8)(2x – 2)

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Review – Solving Quadratics (Factoring)

Factor and solve for the variable.

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Segment in Circles WITH Quadratics

Name Theorem Hypothesis Conclusion

Segment Chord

Theorem

If two chords in a circle

interest, then the

product of the lengths of

the segments of one

chord is equal to the

product of the lengths of

the segments of the

second chord.

Example: Find x

Secant Segment

Theorem

If two secant segments

intersect in the exterior of

a circle, then the

product of the lengths of

the secant segment and

its external secant

segment is equal to the

product of the lengths of

the second secant

segment and its external

secant segment.

Example: Find x

Secant Tangent

Theorem

If a tangent and secant

intersect in the exterior

of a circle, then the

product of the lengths

of the secant segment

and its external secant

segment is equal to the

square of the length of

the tangent segment.

Example: Find x

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GUIDED PRACTICE

(Outside x Outside = Outside x Whole) 1. Solve for x. 2. Solve for x.

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(Outside x Outside = Outside x Whole)

3. Solve for x 4. Solve for x

(Outside x Whole = Outside x Whole)

5. Solve for x 6. Solve for x

(Outside x Whole = Outside x Whole)

7. Solve for x 8. Solve for x

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(Part x Part = Part x Part) 9. Solve for x. 10. Solve for x.

11. Solve for x. 12. Solve for x

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Skills Practice – Segments QUADRATICS

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7. Solve for the measure of x 8. Solve for the measure of x

9. Solve for the measure of x 10. Solve for the measure of x

11. Solve for the measure of x 12. Solve for the measure of x

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Notes - Lengths of Circle Arcs

The distance around a circle is called the Circumference. This can be found by using this equation:

Find the Circumference of the following circles: Leave your answers in terms of π as well as a decimal.

1) 2) 3)

Practice reviewing how to calculate the circumference or radius/diameter of a circle below. Leave your

answers in terms of pi. Find the circumference, radius, or diameter.

A. r = 6 ft B. d = 15 in C. C = 16 cm D. C = 40 m

Circumference

C = 2 r or C = d

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ARC LENGTH

Arc Length is a fraction of the circle’s circumference and is measured in linear units. Arc length can be

calculated using the following proportion or EQUATION

WE DO: Leave in terms of π

b. Find the arc length of 𝐴�̂�

length 𝐴�̂� = 120

2 (4)360

length 𝐴�̂� = 1

83

length 𝐴�̂� = 8

3units ≈ 8.4 𝑢𝑛𝑖𝑡𝑠

YOU DO: Leave in terms of π

b. Finding the length of arc 𝐴�̂�

Length 𝐴�̂� =

Arc Length

=

arc measure of angle arc length

360 circumference (2 r)

Given: P and m APC = 120˚

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360o

Arc Measure Arc Length

Circumference=

ARC LENGTH

WE D0: Finding the length of arc 𝑩�̂�

YOU D0: Finding the length of arc 𝑩�̂�

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Arc Length Guided Practice

Find the arc lengths for problems 2 and 3.

1. Length of arc RS= 2. Length of arc MN = 3. Length of arc AB =

(exact answer) (approx. answer) (exact answer)

4. A circle has a radius of 6 cm. A sector has an arc length of 8.4 cm. The angle at the center of the sector is θ.

Calculate the value of θ.

5. Find the radius of circle N.

6. Find the circumference of circle Q.

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7. A clock has a pendulum 22 centimeters long. If it swings through an angle of 32 degrees, how far does the bottom of the pendulum

travel in one swing?

For questions 8-9, use the figure below:

8) How many degrees does the minute hand move in 15 minutes? 40 minutes? 55 minutes?

9) If the minute hand is 4 inches long, what is the arc length covered by the minute hand in 40

minutes?

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Skills Practice: Calculating Arc Length and Circumference

Use the diagram to find the indicated measure. Leave answers in term of pi.

1. Find the circumference. 2. Find the circumference.

3. Find the radius. Find the indicated measure.

a. The exact radius of a circle with circumference 36 meters

b. The exact diameter of a circle with circumference 29 feet

c. The exact circumference of a circle with diameter 26 inches

d. The exact circumference of a circle with radius 15 centimeters

4. Find the length of AB .

a. b. c.

5. In D shown below, ADC BDC. Find the indicated measure

a. mCB

b. mACB e. mBAC

c. Length of CB

f. Length of ACB

d. Length of ABC

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36

D

E

C

80

120

D

E

6. Find the indicated measure.

a. The radius of circle Q b. Circumference of Q and Radius of Q

Find the perimeter of the region. Round to the nearest hundredth.

7.

8. Birthday Cake A birthday cake is sliced into 8 equal pieces. The arc length of one piece of cake is 6.28

inches as shown. Find the diameter of the cake.

9. Radius = 5 in 10. Find the radius of the circle.

Length of Arc CE = _______ r = ___________

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For #11-13, solve for the requested variable. C is the center of each circle.

11. r = _______ 12. x° = ________ 13. d = ______

14. Circumference = 10 m; Find the arc length of JT = ______

15. The arc length of OP = 10𝜋 inches; 16. The arc length of QT = 22 cm.;

r = _______ d = _______ (to the tenth)

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Skills Practice – Arc Length

Practice: Find the length of each bold arc. Write your answers in terms of π as well as a decimal rounded to one

decimal place.

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Practice: Find the length of each bold arc. Write your answers in terms of π as well as a decimal rounded to one

decimal place.

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STUDY GUIDE

Learning Target #1 – Tangents

1. Is HM tangent to the circle? 2. Is FG tangent to the circle?

3. Find the value of x: 4. Is AB tangent to the circle?

5. Find the length of ?: 6. What is the perimeter?

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Learning Target #1 – Chords

1. What is the length of ST? 2. Find the value of x and y:

3. Find the value of x: 4. mMN

5. Find the value of x: 6. Find the value of x:

y

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Station 2 – Segments Without Quadratics

1. What is the length of NR? 2. What is the value of x? (Part x Part = Part x Part)

3. What is the value of x? 4. What is the value of x? (Outside x Whole = Outside x Whole)

5. What is the value of x? 6. Solve for x.

(Outside x Outside = Outside x Whole)

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Station 2 – Segments With Quadratics

2. What is the length of BC? 2. What is the value of x? (Part x Part = Part x Part)

4. What is the value of WY? 4. What is the value of x? (Outside x Whole = Outside x Whole)

5. What is the value of x? 6. Solve for x.

(Outside x Outside = Outside x Whole)

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Learning Target #3 – Arc Length and Circumference

1. If the radius of circle A is 27, 2. What is the circumference of

what is the circumference? circle P shown below?

C = ____________ C = ____________

3.What is the arc length 4. What is the arc length

5. Given the arc length, find the radius 6. Given the arc length, find the radius

7. What is the length of DE shown below? 8. What is the length of 𝐶𝐷⏜ shown below?

7