Grade 10 math

1. Lesson 1
The Circle and
Its Parts

2. Objectives
At the end of this lesson, the learner should be able to
● accurately determine the different parts of a circle;
● correctly solve for the measure of the arc of a circle;
and
● correctly solve word problems involving the parts of
a circle.

3. Essential Questions
● What are the different parts of a circle?
● How will you solve for the measure of an arc of a circle?

4. Warm Up!
Before we learn the different parts of a circle, let us learn
how to construct figures involving a circle using an online
tool!
(Click on the link to access the exercise.)
“Geometry”. Geogebra. Retrieved 15 April 2019 from
https://www.geogebra.org/geometry

5. Photo place holder
Geogebra!
Let’s construct!

6. Geogebra!
Let’s construct!
Construct the following figures in Geogebra.
1. Construct circle with center A.
2. Plot point B on the circle and construct 𝐴𝐵
3. Draw a line which passes through A and whose
endpoints are on the circle. Name it 𝐶𝐷.
4. Draw a line whose endpoints are on the circle but
does not pass through the center. Name it 𝐸𝐹.

7. Guide Questions
● What kind of lines are 𝐴𝐵, 𝐶𝐷, and 𝐸𝐹?
● Do you think those are the only parts of a circle? Can you
give other parts of a circle?
● Why is it important to determine the different parts of a
circle?

8. Learn about It!
1 Circle
set of all points on a given plane that is equidistant from a fixed point on the
plane called the center of the circle; named based on the letter used to indicate
the center of the circle
Example:
The circle to the right is named circle 𝐶.

9. Learn about It!
2 Radius
distance between the center of the circle and a point on the circle; the plural form
of radius is radii
Example:
The lines 𝑄𝐶, 𝐶𝑅, and 𝑃𝐶 are the radii of
circle 𝐶.

10. Learn about It!
3 Chord
a line segment whose endpoints lie on the circle
Example:
The lines 𝑄𝑅, 𝑃𝑅, and 𝑄𝑃 are chords of
circle 𝐶.

11. Learn about It!
4 Diameter
a chord that passes through the center of the circle; the diameter is also the
longest chord in a circle
Example:
The line 𝑄𝑆 is the diameter of circle 𝐶.

12. Learn about It!
5 Secant
a line that intersects a circle in two points
Example:
The lines 𝑄𝑆 and 𝑄𝑅 are secants of
circle 𝐶.

13. Learn about It!
6 Tangent
a line that intersects a circle at only one point; the point where it intersects the
circle is called the point of tangency
Example:
The line 𝑈𝑉 is tangent to circle 𝐶, and
the point of tangency is point 𝑇.

14. Learn about It!
7 Arc
a portion of a circle formed between two points on the circle
Example:
The arc 𝑃𝑄 is a minor arc, and the arc
𝑃𝑅𝑄 is a major arc.

15. Learn about It!
8 Arc Addition Postulate
The measure of an arc formed by two adjacent arcs of the same circle is equal to
the sum of the measures of the two arcs.
Example:
In the given circle 𝑂, 𝑚 𝑄𝑃 = 90° and
𝑚 𝑃𝑅 = 120°. It follows that
𝑚 𝑄𝑃𝑅 = 𝑚 𝑄𝑃 + 𝑚 𝑃𝑅 = 210°.

16. Learn about It!
9 Central Angle
an angle whose vertex is the center of the circle
Example:
In the figure to the right, ∠𝑄𝑂𝑃 is a
central angle.

17. Learn about It!
10 Central Angle Postulate
The measure of a central angle is equal to the measure of its intercepted arc.
Example:
The measure of the central angle ∠𝑄𝑂𝑃
is 90°. It follows that the measure of the
intercepted arc 𝑄𝑃 is also 90°.

18. Learn about It!
11 Inscribed Angle
an angle whose vertex is on the circle
Example:
In the figure to the right, ∠𝑃𝑅𝑄 is an
inscribed angle.

19. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
a. 𝑂𝐶
b. 𝐴𝐶
c. 𝐴𝐷
d. 𝐴𝐹
e. 𝐶𝐹
f. ∠𝐴𝐷𝐵
g. ∠𝐴𝑂𝐸
h. 𝐸𝐶
i. 𝐴𝐸𝐵
j. 𝐴𝐶

20. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
a. 𝑂𝐶
Solution:
𝑂𝐶 is a line segment connecting
the center 𝑂 to the point of a
circle, which is 𝐶. Thus, 𝑂𝐶 is a
radius.

21. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
b. 𝐴𝐶
Solution:
𝐴𝐶 is a line segment connecting
two points of a circle, which is 𝐴
and 𝐶, and passes through the
center 𝑂. Thus, 𝐴𝐶 is a diameter.

22. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
c. 𝐴𝐷
Solution:
𝐴𝐷 is a line segment connecting
two points of a circle, which is 𝐴
and 𝐷, but does not pass through
the center 𝑂. Thus, 𝐴𝐷 is a chord.

23. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
d. 𝐴𝐹
Solution:
𝐴𝐹 is a line which intersects the
circle at two points, which are 𝐴
and 𝐵. Thus, 𝐴𝐹 is a secant.

24. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
e. 𝐶𝐹
Solution:
𝐶𝐹 is a line which intersects the
circle at one point, which is 𝐶.
Thus, 𝐶𝐹 is a tangent.

25. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
f. ∠𝐴𝐷𝐵
Solution:
∠𝐴𝐷𝐵 is an angle whose vertex lies
on the circle. Thus, ∠𝐴𝐷𝐵 is an
inscribed angle.

26. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
g. ∠𝐴𝑂𝐸
Solution:
∠𝐴𝑂𝐸 is an angle whose vertex is
on the center of the circle. Thus,
∠𝐴𝑂𝐸 is a central angle.

27. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
h. 𝐸𝐶
Solution:
𝐸𝐶 is an arc bounded by the
points 𝐸 and 𝐶. Its measure is less
than 180°. Thus, 𝐸𝐶 is a minor
arc.

28. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
i. 𝐴𝐸𝐵
Solution:
𝐴𝐸𝐵 is an arc bounded by the
points 𝐴 and 𝐵 and passes
through 𝐸. Its measure is more
than 180°. Thus, 𝐴𝐸𝐵 is a major
arc.

29. Try It!
Example 1: Name the following parts of circle 𝑂 given below:
j. 𝐴𝐶
Solution:
𝐴𝐶 is an arc bounded by points 𝐴
and 𝐶. Since 𝐴𝐶 is a diameter, it
follows that 𝐴𝐶 is a semicircle.

30. Try It!
Example 2: In the figure below, 𝑚∠𝐵𝑂𝐶 = 60° and
𝑚∠𝐴𝑂𝐵 = 135°. Find 𝑚 𝐴𝐶.

31. Try It!
Example 2: In the figure below, 𝑚∠𝐵𝑂𝐶 = 60° and
𝑚∠𝐴𝑂𝐵 = 135°. Find 𝑚 𝐴𝐶.
Solution:
We will use the Central Angle Postulate to
determine 𝑚 𝐴𝐶. This postulate states that
the measure of a central angle is equal to
the measure of its intercepted arc.

32. Try It!
Example 2: In the figure below, 𝑚∠𝐵𝑂𝐶 = 60° and
𝑚∠𝐴𝑂𝐵 = 135°. Find 𝑚 𝐴𝐶.
Solution:
Since 𝑚∠𝐵𝑂𝐶 = 60°, it follows that the
measure of its intercepted arc 𝐵𝐶 is also 𝟔𝟎°.
Since 𝑚∠𝐴𝑂𝐵 = 135°, it follows that the
measure of its intercepted arc 𝐴𝐵 is also
𝟏𝟑𝟓°.

33. Try It!
Example 2: In the figure below, 𝑚∠𝐵𝑂𝐶 = 60° and
𝑚∠𝐴𝑂𝐵 = 135°. Find 𝑚 𝐴𝐶.
Solution:
Remember that the measure of the arc
equivalent to an entire circle is 360°. Thus,
we can add the measures of the three arcs
and make it equal to 360°.
𝑚 𝐴𝐵 + 𝑚 𝐵𝐶 + 𝑚 𝐴𝐶 = 360°

34. Try It!
Example 2: In the figure below, 𝑚∠𝐵𝑂𝐶 = 60° and
𝑚∠𝐴𝑂𝐵 = 135°. Find 𝑚 𝐴𝐶.
Solution:
Solving for 𝑚 𝐴𝐶, we will get 𝑚 𝐴𝐶 = 165°.
Therefore, 𝒎 𝑨𝑪 = 𝟏𝟔𝟓°.

35. Let’s Practice!
Individual Practice:
1. Determine all the radii,
diameters, chords, secants,
and tangents of the figure to
the right.

36. Let’s Practice!
Individual Practice:
2. In the figure below, 𝐴𝐶 and 𝐵𝐷 are
diameters intersecting at center 𝑂.
If 𝑚∠𝐴𝑂𝐵 = 55°, find 𝑚 𝐴𝐵, 𝑚 𝐴𝐷,
and 𝑚 𝐴𝐶𝐷.

37. Let’s Practice!
Group Practice: To be done in groups of two to five
Determine the measure of the intercepted arc of the angle
formed by the hour and minute hand of a clock if the current
time is 3:20.

38. Key Points
1 Circle
set of all points on a given plane that is equidistant from a fixed point on the
plane called the center of the circle; named based on the letter used to indicate
the center of the circle
2 Radius
distance between the center of the circle and a point on the circle; the plural form
of radius is radii
3 Chord
a line segment whose endpoints lie on the circle

39. Key Points
4 Diameter
a chord that passes through the center of the circle; the diameter is also the
longest chord in a circle
5 Secant
a line that intersects a circle in two points
6 Tangent
a line that intersects a circle at only one point; the point where it intersects the
circle is called the point of tangency

40. Key Points
7 Arc
a portion of a circle formed between two points on the circle
8 Arc Addition Postulate
The measure of an arc formed by two adjacent arcs of the same circle is equal to
the sum of the measures of the two arcs.
9 Central Angle
an angle whose vertex is the center of the circle

41. Key Points
10 Central Angle Postulate
The measure of a central angle is equal to the measure of its intercepted arc.
11 Inscribed Angle
an angle whose vertex is on the circle

42. Synthesis
● What are the different parts of a circle?
● Why is it important to determine the different parts of a
circle in solving for the measure of its arcs?
● How are chords and radii related?