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Math 10 Q2 Wk5 Illustrates Secant Tangent Sector and
Segment of a Circle Theorems on Secants Tangent
BSED-Mathematics (Agusan del Sur State College of Agriculture and Technology)
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Department of Education
Region III
DIVISION OF MABALACAT CITY
Grade & Section:
Name:
School:
Date:
LEARNING ACTIVITY SHEETS
Mathematics 10 (Q2 – Wk-5)
Illustrates Secant, Tangent, Sector and Segment of a Circle
Theorems on Secants, Tangents and Segments
I. Introduction
These Learning Activity Sheets will help you to have a clearer idea of
what are secant, tangent, sector and segment of a circle. Likewise, you will
understand the theorems involving tangent and secant segment. As you go over
the activities, you will be able to demonstrate understanding of the lesson where
tangent, secant segments and sectors are applied.
II. Learning Competency
Illustrate secant, tangent, sector, and segment of a circle (M10GE-IIe-1)
Proves theorems on secants, tangents, and segments (M10GE- lle-f-1)
III. Objectives
After going through these learning activity sheets, you are expected to:
1. define and illustrate secant, tangent, sector, and segment of a circle
2. find the area of a sector and segment of a circle;
3. determine the measure of the angles formed by secant and
tangent lines;
4. find the length of the unknown segment in a circle using the theorems
related to chord, tangent and secant; and
5. prove the theorems on secant segments, tangent segment, and external
secant segments.
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IV.
Discussions
Secant and Tangent of a Circle
The secant of a circle is a straight line that intersects a circle in exactly two
points.
F
B
A
In the figure above ⃡� � intersects circle F in A and B. Since the line intersects
the circle at exactly two points, then ⃡� � is the secant of circle F.
The tangent of a circle is a straight line that intersects a circle at exactly one
point. The point where the circle and the tangent line intersects is called the
point of tangency.
F
V
A
P
In the figure above, ⃡� � intersects circle F at point A. Since ⃡� � intersects the
circle at exactly one point, then ⃡� � is tangent to circle F and A is the point
of tangency.
Sector of a Circle
A sector of a circle is a portion, or a region bounded by two radii and either by
the intercepted minor arc or major arc.
2
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Area of a Sector of a Circle
The area or a sector is a portion of the area of the entire circle.
A = �× ��2,
360
where
A = area of the sector
C = measure of the central angle r = radius of the circle
Since the measure of the central angle is equal to the measure of the intercepted
arc, the formula in getting the area of the sector if the measure of the intercepted
arc is given can be written as
A = �36
× ��2,
0
where
A = area of the sector
m = measure of the intercepted arc
r = radius of the circle
Example/s:
1. Find the area of a sector with a central angle of 70o and a radius of
8 cm.
Solution:
A=
�
36
0
× ��2 =
70
( � ∙ 82 ) = 12.44 � cm2
360
The area of the sector is 12.44 � cm2
2. The measure of the arc of a sector of a circle is 65 0. The length of the
radius is 9 cm. Find the area of the sector.
Solution:
A=
�
36
0
× ��2 =
65
( � ∙ 92 ) = 14.63 � cm2
360
The area of the sector is 14.63 � cm2
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Segment of the Circle
A segment of the circle is the region bounded by a chord and the minor arc. It
is a region of the area of the circle cut off by a chord.
segment
The area of the segment is determined by subtracting the area of the triangle
formed by the chord and the radii from the area of the sector.
Asegment = Asector - Atriangle
Example/s:
1. Find the area of the segment.
B
600
8 cm
A
C
Solution: Since the triangle formed is equailateral where the area of equilateral
triangle is given by (�2 √3)
4
Asegment = Asector - Atriangle
Asegment = ( �
360
× ��2) – (�2 √3)
4
60
= ( 36 × � ∙ 82) – ( 82 √3 )
0
4
= (10.6.7�) – (27.71)
= 33.51 – 27.71
= 5.8 cm2
The area of the segment is 5.8 cm2
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2. Find the area of the segment.
D
1200
E
9 cm
F
Solution: To find the area of ∆DEF, draw a line segment from point E
perpendicular to chord DF to form 30o, 60o and 90o triangle. In 30o, 60o
and 90o triangle, the length of the hypotenuse is twice the length of the
shorter leg and the length of the longer leg is √3 times the shorter leg.
Using ∆EGF, where ∠F =30o , ∠1= 60o and ∠G=90o.
EG = ½ (EF)
= ½ (9) = 4.5
D
FG = EF (√3)
= EF (√3)
0
120
∆EGF ≈ ∆EGD
G
= (4.5) (√3)=7.79
E
1
9 cm
Asegment = (
F
�
360
× ��2) – Area of ∆DEF
120
2
= ( 360 × � ∙ 9 ) – (Area of ∆EGF + Area of ∆EGD)
2
120
= ( 360 × � ∙ 9 ) – 2(Area of ∆EGF)
120
= ( 360
× � ∙ 92) – 2(1)(4.5)(7.79)
2
= (27�) – (35.06)
= 84.82 – 35.06
= 49.76 cm2
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Theorems on Angles Formed by Tangents and Secants
1. If two secants intersect in the exterior of a circle, then the measure of the
angle formed is one-half the positive difference of the measures of the
intercepted arcs.
Example: In the figure m DC = 1300 and m IO = 400. Find mDVC .
D
I

Solution:
V
O
C
1
mDVC =(m ID – m VC)
2
s intersecting outside the circle at point V are ID and OC. The two intercepted arcs of DVC
are DC and IO.
mDVC = 1(130 – 40)
2
1
=(90)
2
mDVC = 450
2. If a secant and a tangent intersect in the exterior of a circle, then the
measure of the angle formed is one-half the positive difference of the
measures of the intercepted arc.
Example: If m POH = 1780 and m EH = 540, what is mHSP ?
P

O
E
S
H
Solution:
1
mHSP =(m POH – m EH)
2
s a secant and SH is a tangent intersecting outside a circle at point S. The two intercepted arcs of
mHSP = 1(178 – 54)
and EH.
2
1
=(124)
2
mHSP = 620
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3. If two tangents intersect in the exterior of a circle, then the measure of the
angle formed is one-half the positive difference of the measures of the
intercepted arcs.
Example: If m ACE = 2640 and m AE = 1120, what is mAFE ?
A

F
C
E
1
mAFE =(m ACE – m AE)
2
ts intersecting outside the circle are FA and FE at point F. The two intercepted arcs of 1
AFE are ACE and AE.
mAFE = (264 – 112)
2
Solution:
1
=(152)
2
mAFE = 760
4. If two secants intersect in the interior of a circle, then the measure of the angle
formed is one half the sum of the measures of the arcs intercepted by the angle
and its vertical angle.
Example: In the figure, m FA= 400, and m HT = 500. Find m 1.
F
A
1
2 I
H
T
Solution:
1
m
=(mFA + mHT)
2 intercepted arcs of 2 are FH and A
nside the circle areFT and HA. The two intercepted arcs of 1 are FA and HT while the
1
m
=(40 + 50)
2
m
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1
=(90)
2
= 450
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5. If a secant and a tangent intersect at the point of tangency, then the
measure of each angle formed is one-half the measure of its intercepted arc.
Example: In the figure, m CAE is 2100, what is mCER ?
C
A

R
E
S
Solution:
mCER = 1 mCAE
2
E is the secant and RS is a tangent intersecting at E, the point of tangency. The intercepted
arc of
hile the intercepted arc
mCER = 1(210)
2
mCER = 1050
Theorem on Two Intersecting Chords
If two chords intersect in a circle, then the product of the measures of the
segments of one chord is equal to the product of the measures of the
E
other chords
R
O
S
RO▪OS= TO▪OE
T
Example 1: Solve for x in the given figure.
Solution:
4(x) = 6 (8)
4x = 48
4�
48
4= 4
x = 12
4
6
8
x
C
Example 2: Two chords �� and ��
intersecting at point M. If MA = (4x – 2)
cm, MB = 6cm, MC = (3x + 3) cm and MD
= 5cm. What is the measurement of
segment AB?
M
B
A
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D
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Solution:
MA (MB) = MC (MD) (4x – 2)(6) = (3x + 3)(5) 24x – 12 = 15x + 15
24x – 15x = 15 + 12
by Distributive Property of Multiplication
9x = 27
(4x – 2)(6) = (3x + 3)(5)
x = 3 Since AB = MA + MB
Then, AB = (4x – 2) + (6) AB = 4x + 4
If x = 3, then AB = 4(3) +6
AB = 12 + 6
AB = 18
by Transposition Method
24x – 12 = 15x + 15
divide both sided by 9 to eliminate the coefficient of the variable x
9� =
9
27
Based on the solution, the measurement of segment AB is 18 cm.
Theorems on Secant Segments, Tangent Segments, and
External Secant Segments
1. If two secant segments are drawn to a circle from an exterior point, then
the product of the lengths of one secant segment and its external secant segment
is equal to the product of the lengths of other secant segment and its external
secant segment.
SI and EI are the secant segments.
MI and LI are the external secant segments.
SI  MI = EI  LI
Example 1: Find the length of the unknown segment in the following figures. a.
T
4
3
I
E
K
8

x
S
Solution:KT  IT= ST  ET
(8 + 4) (8) = (x + 3) (3)
12( 8 ) = 3x + 9
96 = 3x + 9
96- 9 = 3x
87 = 3x
87
=
3�
33
x = 29
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A
Solution:AR  WR= SR. DR
(x + 7) (x) = (6 + 4) (6)
x2 + 7x = 10 (6) x2 + 7x = 60
x2 + 7x – 60 = 0Factor (x +12)(x -5) = 0
x=5
7
W
b.
x

S
4
D
R
6
Example 2: MA and MX are Secant segments, prove that MA  MB = MX  MY.
1
2
Proof:
Statements
1. Secant segments MA and MX
2. Draw BX and AY
3. ∠
� ≅∠
4. ∠1 ≅ ∠2
5. ∆
���
��
∼∆
���
��
6. �� =��
7. MA  MB = MX MY
Reasons
1. Given
2. Two points determine a line
3. Reflexive property
4. Inscribed angles with the same
intercepted arcs are congruent.
5. AA Similarity Theorem
6. Corresponding sides of similar
triangles are proportional.
7. Fundamental Law of Proportion
2. If a tangent segment and a secant segment are drawn to a circle from an
exterior point, then the square of the length of the tangent segment is equal to
the product of the lengths of the secant segment and its external
secant segment. O NP is a secant segmentPand OP is a tangent segment drawn to the circle from the ex
(OP)2 = NP  EP

E
N
Example 1: Find the length of the unknown segment in the following figures.
S
Solution:(SP)2= PT  OT
x
a.
(x)2 = ( 9 + 16) ( 9 )
T
x2 = 25 ( 9)

O 9
x2 = 225
16
P
√�2=
√225
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x = 15
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b.
86
x O
R
Solution: MR  OR = (RE)2
(x + 6) (x) = (4)2 x2 + 6x = 16
x2 + 6x – 16 = 0Factor
(x + 8)(x – 2) = 0;x = 2
M

4
E
Example 2: VL is a secant segment and VE is a tangent segment, prove that
(VE)2 = VO  VL
Proof:
Statements
Reasons
1. VL is a secant segment and
VE is a tangent segment
2. Draw LE and EO
3. ∠� ≅ ∠
1
4. �∠���
= �̂�; ∠
�∠���
=
̂��
2
2
5.∠VEO≅∠VLE
6. ∆��� ∼ ∆���
��
��
7. �� =��
8. (VE)2 = VO  VL
1. Given
1
2. Two points determine a line
3. Reflexive property
4. measures of inscribed angles is
one-half the measure of its
intercepted arc.
5. Definition of congruent angles
6. AA Similarity Theorem
7. Corresponding sides of similar
triangles are proportional.
8. Fundamental Law of Proportion
V. Activities
Activity 2
A. Find the area of the sector and the segment given the radius and the
measure of the central angle.
1 – 2. r = 15 cm, m = 1200
3 – 4. r = 16 cm, m = 600
5 – 6. r = 11 cm, m = 900
7 – 10. Find the area of the shaded regions. 7
– 8.
B
9 – 10.
1200
600
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A
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20 cm
C
Activity 2
Solve the given problem below.
Kelly bought a pizza and got her a portion of it and the remaining was
given to her siblings. If the portion of pizza she got forms an angle of 75 0 and
the pizza has a diameter of 12 inches.
1-2.
Draw the figure that best illustrates the problem.
3. What is the area of the portion of the pizza that was given to
her siblings?
4. What is the area of the pizza slice that Kelly has?
5. If Kelly has 5 siblings and the next time, she will buy a pizza all of them
will get the same share, what would be the area of the pizza slice if they
will still buy a 12 inches pizza?
Activity 3
A. Use the figure below to find the measures of the following:
1. m LS = 1000 , m DR = 200, mRMD =
2. mRMD = 500, m DR = 300, m LS =
3. m LS = 1200, mRMD = 300, m RD =
4. m LS= 1700, m DR = 650, m RMD =
5. mRMD = 520, m LR = 780, m LS = 1380, m RD= 340, m DS=
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B. Determine the measure of the angles formed by secant and tangent lines.
Write the corresponding word on the blank above the answer to find out the
message.
If m STR = 1600 and m AS = 700.
Find AMS .
THEM.
If m RUE = 2380, findRTE .
CREATE


If m ST = 740, find2 .
If m IXF = 2250, find 1.
HAPPEN.
OPPORTUNITIES
5.
6.
If m AR = 1300 and m SC = 1500,
find 1.
YOU
If m WR = 800, findWOR .
DON’T
What is the message?
1430
1000
112.50
1400
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580
450
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Activity 4
Find the length of the unknown segment (x) in each of the following figures.
x
1.
2.
3.
x
x6
x+8
6
3

6


9
9
9
4.
14
5.
x
7 
10
6

6.
12
8
6
5
7.
x
8.
5
x
6

x
9.
4


x
x
9
8
8
Activity 5
Complete the table below. NF is tangent and ND is secant to circle S. Prove
that (ND)2 = NI  NF .
D
N
I
Proof:
Statements
1.
2.
3. ∠
4.
≅∠
5. ∆
6. � � =
��
7. (ND)2 =
∼

F
Reasons
1. Given
2. Two points determine a line
3.
4. Angles with the same intercepted
arc are congruent
5.
6. Corresponding sides of similar
triangles are
.
7.
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8
12
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VI.
Assessment
Read each question carefully then encircle the letter of the best answer from
the given choices.
1. It is a line that intersects the circle with exactly two points.
A. Secant
C. Chord
B. Tangent
D. Diameter
2. It is a line that intersects the circle with a point.
A. Secant
C. Chord
B. Tangent
D. Diameter
3. What is the area of the sector of a given circle whose radius is 4 inches
and the measure of the central angle is 600?
A. 17.55 inches2
C. 12.38 inches2
B. 16.5 inches2
D. 8.38 inches2
4. What is the area of the segment in item no. 3?
A. 1.45 inches2
C. 3.57 inches2
B. 2.15 inches2
D. 4.62 inches2
5 – 6. In the figure below, which are the secant and the tangent,respectively?
C
D
A
F
BE
A. ⃡⃡�
⃡
⃡
� ,�
�
C. �
� , ⃡�
B. �⃡ � , ⃡� �
D. ⃡� � , ⃡� �
7. In figure above, which is the point of tangency?
A. Point A
B. Point B
C. Point C
�
D. Point D
8. Find the value of x in the figure below.
8
10
C. 14
B. 12
D. 16
A.
9. Two tangents to a circle form an angle of 800. What is the measure of the
smaller intercepted arc?
A. 1500
B. 1200
C. 1000
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D. 2100
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10. Complete the given statement. If the tangent and secant segments intersect
outside the circle, then the
of the tangent segment is
equal to the product of the outer secant segment and the secant segment.
A. cube
B. square root
C. square
0
11. In the figure, if m BST = 245 , what is mTSB ?
D.twice
B
A.
2450
C. 1200
B.
1100
D. 122.50

T
E
S
12. In the figure, if m DEN = 2500, what is the measure of DON ?
D
A.700
E

C. 1400
0
O B. 200
D. 1000
N
For numbers 13 – 15, refer to the given figures below.
6

15
x


10
FIGURE 1
FIGURE 3
FIGURE 2
13. What is the value of x in Figure 1?
A. 80 cm
B. 18 cm
C. 12 cm
D. 144 cm
14. In figure 2, if VI = 8 in, ID= 22 in and VO= 10 in, which of the following is
the measurement of segment OC?
A. 14 in
B. 16 in
C. 24 in
D. 36 in
15. In figure 3, what is the value of x?
A. 8
B. 6
C. 12
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D. 4
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VII.
Reflections
A. Use the Venn Diagram to compare and contrast secant and tangent. In the
outer circles, right their description/s that are different and in the inner
circle, write description/s that are the same.
RUBRIC
5 pts. – At least one description was given
for outer circles and inner circle.
3 pts. – At least one description was given
for each circles.
1 pt. – At least one description was given in
any of the circles.
0 – No description was given.
secan
tange
B. Follow the given instructions below to illustrate what is being described.
6. Draw a circle and name it as Circle A.
7. Draw secant �⃡ �
in anywhere on the circle A.
8. Draw tangent ⃡� �
with point of tangency B.
RUBRIC
5 pts. – All instructions and the
right figure were illustrated.
3 pts. – Exactly two of the
instructions were followed 1
pt. – Exactly one of the
instructions was followed
0 – No figure was illustrated.
C. Fill in the blanks with the correct answers based on what you
have learned.
1. If two
intersect in a circle, then the product of the measures of
the segments of one chord is equal to the
of the measures of
the other chords.
2. If a tangent segment and a secant segment are drawn to a circle from
an exterior point, then the
of the length of the tangent segment
is equal to the product of the lengths of the secant segment and its
.
3. If two secants intersect in the interior of a circle, then the measure of
the angle formed is one half the
of the measures of the arcs
intercepted by the angle and its
.
4. If two secants intersect in the
of a circle, then the measure of
the angle formed is one-half the positive difference of the measures of
the
.
5. If two tangents intersect in the exterior of a circle, then the measure of the
angle formed is
the positive
of the measures of the
intercepted arcs.
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VII. References
DepEd-Bureau of Secondary Education Curriculum Development Division.
“Grade 10 Mathematics Learner’s Module”. Pasig City:
Dominguez, Ricardo D., Federizo, Rogel H. 2015.
Generation” Quezon City: Bright House Publishing
“Math for Today’s
Orines, Fernando, et.al. 2019. “Next Century Mathematics: Second Edition”.
Phoeniz Publishing House.
Ponsones, Rigor B.,Ocampo , Shirlee R., Tresvaless, Regina M., 2013. “Math
Ideas and Life Application” Quezon City: Abiva Publishing House, Inc.
Ulpina, Jisela, N.,Razon, Lerida.2015. “Math Builders”. JO-ES Publishing
House Incorporated
Orines, Fernando, et.al. 2019. “Next Century Mathematics: Second Edition”.
Phoeniz Publishing House.
Larry
Schmidt.
Retrieved
https://www.youtube.com/watch?v=Gqz9WUsKh80
from:
https://www.superprof.co.uk/resources/academic/maths/geometry/plane/
circle-word-problems.html#chapter_solution-of-exercise-1
https://byjus.com/maths/tangent-of-a-circle/
https://byjus.com/maths/secant-of-a-circle/
https://www.cliffsnotes.com/study-guides/geometry/circles/segments-ofchords-secants-tangents
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2003.fall/Nichols/6690/W
ebpage/Day%209.htm
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VIII. Answer Key
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IX. Development Team
Development Team of the Learning Activity Sheets
Writers: Frances Ann V. Pamintuan
Maricel O. Dayrit
Editor: Angelito S. Cabrera Reviewer:
Illustrator:
Layout Artist:
Management Team: Engr. Edgard C. Domingo, PhD,CESO V
Leandro C. Canlas, PhD, CESE
Elizabeth O. Latorilla, PhD
Sonny N. De Guzman, EdD
Elizabeth C. Miguel, EdD
For inquiries or feedback, please write or call:
Department of Education – Division of Mabalacat
P. Burgos St., Poblacion, Mabalacat City, Pampanga Telefax: (045) 331-8143
E-mail Address:
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