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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