3. • Have you imagined yourself pushing a cart or riding in a
bus having wheels that are not round?
• Do you think you can move heavy objects from one place
to another easily or travel distant places as fast as you can?
• What difficulty do you think would you experience without
circles?
• Have you ever thought of the importance of circles in the
field of transportation, industries, sports, navigation,
carpentry, and in your daily life?
4. Arrange the following jumbled letters to form
a word related to circle
•USIDRA
•RADIUS
•RETEMIAD
•DIAMETER
•ORCHD
•CHORD
6. • ISNIRCBED LESANG
• INSCRIBED ANGLES
• ELCRICSEMI
• SEMICIRCLE
• TNEGNAT NILE
• TANGENT LINE
• ESNACT ILEN
• SECANT LINE
7. CHORDS, ARCS, and CENTRAL ANGLES
•Activity 1: Know My Terms and
Conditions
•Use the figures to identify and
name the following terms related to
circle.
10. 6.a major arc
7.a central angle
8.a tangent line
9.a secant line
10.an inscribed angle
11. • Radius – half the measure of the diameter
• Diameter- twice the measure of the radius and it is the longest
chord
• Chord- segment joining any two points on the circle
• Semi-circle- an arc measuring one-half the circumference of a
circle
• Minor arc- an arc of a circle measures less than the semi circle
• Major arc- an arc of a circle that measures greater than the semi
circle
12. • Central angle- an angle whose vertex is the center of
the circle and with two radii as its sides
• Inscribed angle- an angle whose vertex is on the circle
and whose sides contain chords of the circle
• Tangent line- a line segment that pass on the circle at
exactly one point.
• Secant line- a line segment that pass on the circle at
exactly two points
13. •How do you describe a radius, diameter, chord of
a circle?
•How about the semicircle, minor arc, major arc,
tangent line and secant line of a circle? Inscribed
and central angle?
•Write your answer in your notebook.
14. ARCS AND CENTRAL ANGLES
•What is the relation of the
measures of central angles and
its intercepted arcs?
15. Activity 2: Measure Me and You will See…
1. What is the measure of each of the following
angles in Figure 1? Use a protractor.
a. <TOP d. <ROS
105 90
b. <POQ e. <SOT
75 30
c. <QOR
60
16. 2. In figure 2, AF, AB, AC, AD, and AE are radii of circle A. What is
the measure of each of the following angles? Use a protractor.
a. <FAB
105 d. <EAD
90
b. <BAC
75 e. <EAF
c. <CAD 30
60
17. 3. How do you describe the angles in each figure?
The angles have a common vertex
4. What is the sum of the measures of <TOP, <POQ,
<QOR, <ROS, and <SOT in figure 1?
360
How about the sum of the measures of <FAB, <BAC,
<CAD, <EAD, and <EAF in figure 2?
360
18. 5. In figure 1, what is the sum of the measures of the
angles formed by the coplanar rays with a common
vertex but with no common interior points? 360
6. In figure 2, what is the sum of the measures of the
angles formed by the radii of a circle with no common
interior points? 360
7. In figure 2, what is the intercepted arc of <FAB, how
about <BAC, <CAD, <EAD, <EAF, ? Complete the table
below.
19. CENTRAL ANGLE MEASURE INTERCEPTED ARC
<FAB 105 FB
<BAC 75 BC
<CAD 60 CD
<EAD 90 ED
<EAF 30 EF
20. 8. What do you think is the sum of the measures of the
intercepted arcs of <FAB, <BAC, <CAD, <EAD, and <EAF?
Why? 360, the measure of the central angle is equal to the
measure of its intercepted arc.
9. What can you say about the sum of the measures of the
central angles and the sum of the measures of their
corresponding intercepted arc? equal
10. What is the relationship between the measures of the
central angle and its intercepted arc? The measure of the
central angle is equal to the measure of its intercepted arc.
21. Identify and Name Me
•Use the circle below to identify the following.
•1. semicircles in the figure
•2. minor arcs and their corresponding major
arcs
•3. central angles
22. CENTRAL ANGLE AND ARCS
• Sum of Central angles
The sum of the measure of central angles of a circle with no
common interior points is 360.
Degree measure of an arc
The degree measure of a minor arc is the measure of the central
angles which intercepts the arc.
The degree measure of a major arc is equal to 360 minus the
measure of the minor arc with the same end points.
23. Congruent circles and congruent arcs
Congruent circles
Are circles with congruent radii.
Congruent arcs
Are arcs of the same circle or of congruent circles with equal
measures.
24. Theorems on Central Angles, Arcs, and Chords
1. In a circle or in congruent circles, two minor arcs are congruent if
and only if their corresponding central angles are congruent.
2. In a circle or in congruent, two minor arcs are congruent if and only
if their corresponding chords are congruent.
3. In a circle, a diameter bisects a chord and an arc with the same
endpoints if and only if it is perpendicular to the chord.
25. Find my degree measure
• In circle A below, m<LAM = 42, m<HAG=30, and <KAH is a
right angle. Find the following measure of an angle or an arc,
and explain how you arrive at your answer.
• 1. m<LAK
• 2. m<JAK
• 3. m<LAJ
• 4. m<JAH
• 5. m<KAM
27. Activity: Find Me
•In the figure, JI and ON are diameters of circle S. Use
the figure and the given information to answer the
following.
•Which angles are congruent? Why?
•If m<JSN=113, find:
•m<ISO
•m<NSI
•m<JSO
28. •Is OJ=IN? How about JN and OI? Justify your
answer.
•Which minor arcs are congruent? Explain your
answer.
•If m<JSO=67, find the measure of:
•arc JO
•arc JN
30. •Which arc are semicircles? Why?
•Activity: Get My Length
•In the circle M, BD=3, KM=6, and
KP=2squaroot of 7. Use the figure and
the given information to find each
measure.
33. Sector of a circle
•A sector of a circle is the region bounded by two
radii and their intercepted arc. To find the area
of the sector of a circle , get the product of
measure of the central angle/360 and the area
of the circle.
Example; Find the area of the sector of radius 6
cm and its central angle is 70 degrees.
34. Segment of a circle
•A segment of a circle is the region bounded
by an arc and its chord.
•To find the area of the shaded segment,
subtract the area of a triangle from the area
of a sector.
•Area of a segment=Area of a sector-Area of
a Triangle
35. ARC LENGTH
•The length of an arc can be determine by
using the proportion;
•A/360=l/2(3.14)(r),
•Where A-degree measure of the arc
• r-radius of the circle
• l-arc length
36. Arc Length
•Example: If <BEA=90 degrees, and the
radius is 9cm, what is the length of arc
intercepted by the angle.
38. INSCRIBED ANGLES AND ITS INTERCEPTED
ARC
•An inscribed angle is an angle whose
vertex is on a circle and whose sides
contain chords of a circle.
•The arc that lies in the interior of an
inscribed angle and has endpoints on
the angle is called intercepted arc of the
angle.
39. •The measure of the inscribed angle is equal
to one-half of its intercepted arc.
40. Theorems on Inscribed Angles
•1.) If an angle is inscribed in a circle, then
the measure of the angle equals one-half
the measure of its intercepted arc( or the
measure of the intercepted arc is twice the
measure of the inscribed angle)
41. •2.) If two inscribed angles of a circle,
intercept a congruent arcs or the same
arc, then the angles are congruent.
42. •3.) If an inscribed angle of a circle
intercepts a semicircle, then the angle is
a right angle.
43. •4.) If a quadrilateral is inscribed in a
circle, then its opposite angles are
supplementary.
44. Activity
•In the figure, CE and LA are diameters of circle N.
•1.) Name all inscribed angles in the figure.
•2.) Which inscribed angle intercept the following
arcs?
•a.) arc CL
•b.) arc AE
•c.) arc LE
•d.) arc AC
45. •3.) If mLE=124, What is the measure of the
following angles?
•a.) <1
•b.) <2
•c.) <3
•d.) <4
47. •4.) If m<1=26, what is the measure of
each of the following arcs?
•a.) arc CL
•b.) arc AC
•c.) arc AE
•d.) arc LE
48. Activity
•In circle F, AB, BC, CD, BD, and AC are chords.
•1.) Which inscribed angles are congruent?
•Explain your answer.
•2.) If < CBD = 54, what is the measure of arc CD?
•3.) If m arc AB = 96, What is the measure of
<ACB?
49. •4.) If <ABD = 5x + 3 and <DCA = 4x +
10, find:
•a.) the value of x
•b.) m<ABD
•c.) m<DCA
•d.) m arc AD
50. •5.) If m<BDC = 6x – 4 and m arc BC=10x + 2,
find:
•a.) the value of x
•b.) m<BDC
•c.) m arc BC
•d.) m<BAC
51. •5.The Tangent-Secant Theorem
• Given an angle with its vertex on a circle,
formed by a secant ray and a tangent ray, the
measure of the angle is half the measure of the
intercepted arc.
•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 measure of
intercepted arcs.
52. Activity: Inscribed, Intercept, then measure
•In the figure CE and LA are diameters of
circle N. Use the figure to answer the
following.
61. Theorems on Angles Formed by Tangents and
Secants
• If a secant and a tangent line 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.
• 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 measure of the intercepted
arcs.
62. • If two secants intersect in the interior of a circle, then
the measure of an angle formed is one-half the sum of
the measures of the arcs intercepted by the angle and
its vertical angle.
• 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.
63. •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.
67. Theorems on Secant Segments, Tangent
Segments, and External Secant Segments
• If two secant segments are drawn to a circle from 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 the other secant segment and its external
secant segment.
• 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.
68. Theorem on Two Intersecting Chords
• If two chords of a circle intersect, then the product of
the measures of the segments of one chord is equal to
the product of the measures of the segments of the
other chord.
69. Activity 6: Find My Length
•Find the length of the unknown segment(x)
in each of the following figures.
70. Answer the following questions;
•How did you find the length of the
unknown segment?
•What geometric relationships or
theorems did you apply to come up with
your answer?
