Properties of Circles

Radii, Inscribed Angles,

Chord

Circle Theorems

Part 1

Learning Outcome

Understand and apply

circle theorems

- It is an angle formed by two

radii that joins the end of a
chord to the center of a circle.

- INTERCEPTED ARC – an arc

formed when one of two
chords or line segments cut
across a circle and meet at a
common point called the
vertex.

FORMULA:
The central angle = the measure

of the intercepted arc

𝑥°
𝑥°

- Segment AD is a diameter.

Find the values of x, y, z in the
figure.

𝑚∠1 = 1

2𝑚 ෢
𝐴𝐵

- Is an angle whose vertex is on

the circle and whose sides
contain chords of the circle.

- The measure of an inscribed

angle of a circle is equal to
one-half the measure of its
intercepted arc.

2) Inscribed Angle

B

A

1

Triangle ABC is inscribed in

circle O, find:

a) 𝑚 ෢
𝐴𝐶

b) 𝑚∠𝐴
c) 𝑚∠𝐶
d) 𝑚 ෢
𝐴𝐵

Example:

B

C

A

𝟕𝟎°
10𝟎°

- If two inscribed angles
of a circle intercepted
the same arc then the
angles are congruent.

3) Two Inscribed angles

What is the

measurement of angles

D and C?

Example:

B

A

C

D

(7𝑥 − 8)°

(5𝑦 − 3)°

(4𝑦 + 7)°

(5𝑥)°

- The angle in a

semicircle subtended by

the diameter

is a right angle and gives

a 90° angle.

4) Angle in a semicircle

∠𝑎 + ∠𝑏 = 180°

- Is a quadrilateral with all the

four vertices on the

circumference of a circle.

- Opposite angles in a cyclic

quadrilateral are
supplementary
(sum to 180°)

5) Cyclic Quadrilateral

Work out the size of each
marked angle with a letter.

Example:

94°

f

h
g

- The perpendicular

bisector of a chord goes
through the center of a

circle

6) Chord of a circle

In the diagram, M is the
midpoint of the chord AB,

and PQ is the perpendicular

bisector of AB. AB = 8 cm
and MQ = 2cm. Find the

radius of the circle.

Example:

𝑚∠1 =

1
2(𝑚 ෢

𝐷𝐶 + 𝑚 ෢
𝐴𝐵)

𝑚∠2 = 1

2 (𝑚 ෢

𝐴𝐷 + 𝑚 ෢
𝐵𝐶)

- If two chords intersect
inside a circle, then the
measure of each angle is

half the sum of the
intercepted arcs.

7) Two chords intersect

1

2

A

B

C

D

𝑚∠1 = 1

2 𝑚 ෢
𝐴𝐵 𝑎𝑛𝑑 𝑚∠2 = 1

2 𝑚෣
𝐵𝐶𝐴

- If a tangent and a chord
intersect at a point on a

circle, then the measure of
each angle formed is one

half the measure of its

intercepted arc.

8) Chord and tangent

C

B

A

1

2

- The angle between a

tangent and a chord at the

point of contact is equal to the
angle subtended by the chord

in the opposite segment

of the circle

9) Alternate Segment Theorem

- The angle between a

tangent and a chord at the

point of contact is equal to the
angle subtended by the chord

in the opposite segment

of the circle

Example:

i
j

k

35°

Tangent and Secant Lines

Circle Theorems

Part 2

Tangent at a Point

- A circle is itself a curve. There,

at each point on the
circumference of a circle, we
can draw a line called a
TANGENT, to touch the circle
at that point.

- A tangent to a circle is

perpendicular to the radius
drawn from the point of
contact.

Example:

Tangent – Secant Theorems

ANGLES OUTSIDE THE CIRCLE THEOREM

- If a tangent and secant, two tangents, or two secants intersect
outside a circle, then the measure of the angle formed is one half
the difference of the intercepted arcs.

Example:

265



95
C

B

A

Find angle

ACB

Example:

Arc Length

and Area of

Sector

- is a portion of the circumference of a
circle. It can be measured in degrees
or radians to find its length in linear
units.

FORMULA: Arc Length =

𝒙°

𝟑𝟔𝟎°∙

𝟐𝝅𝒓

- The radius of a

circle is 15 cm. If
an arc APB
subtends an angle
of 120° at center
O, finds its length.

Example:

- Sectors of a circle are formed by the
two radii of a central angle and the arc
between their endpoints on the circle.

FORMULA: Area of Sector =

𝒙°

𝟑𝟔𝟎°∙

𝝅𝒓𝟐

- In the diagram, OAPB is

a sector of the circle with
center O and radius of
14 cm. Given that reflex
∠𝐴𝑂𝐵 = 305°, find the
area of the sector OAPB.

Example: