Mathematics presentation.

1. MATHEMATICS 10
Illustrating Tangents, Secants,
Segments, and Sectors
Of a Circle

2. TAKE NOTE OF THESE:
• A tangent is a line that intersects a circle at only one point
and the point of intersection is called the point of tangency.
• A secant is a straight line drawn from the center of a circle
arc to a tangent drawn from the other end of the arc.
• The area of a segment can be determined by subtracting
the area of the triangle formed by the chord and the radii from
the area of the corresponding sector.
• A segment of a circle is a region bounded by a chord and
the arc subtended by the chord. It is a portion of the area of a
circle cut off by a chord.

3. TANGENT
A point of tangency is the point of
intersection between a circle and the
line tangent to a circle. A line is said to
be tangent to a circle (or curve) if it
touches the circle at exactly one point. If
a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency.

4. TWO CIRCLES IN A PLANE MAY HAVE A
COMMON TANGENT.
Line m is a common
external tangent to circles
O and P.
Line n is a common
internal tangent to circles
O and P.

5. Look at Circle E.
AD and BC are secants
intersecting at point O in the
interior of the circle, and
which form vertical angles,
∠AOB and ∠COD, ∠AOC
and ∠BOD.

6. In the next figure, AO and BO are
secants intersecting at point O on the
circle, and which form an inscribed
angle ∠AOB.
∠AOB is inscribed in AOB, and
intercepts AB.
AO and BO are secants intersecting at
point O in the exterior of the circle, and
which form an angle, ∠AOB.
∠AOB intercepts minor arcs AB and
CD.

7. “THEOREMS ON
ANGLES FORMED
BY TANGENTS AND
SECANTS”
^^

8. Theorem 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 intercepted arc.
m∠𝐴𝑂𝐵=
1
2
(m𝐴𝐵 −m𝐶𝐷)
For example, if m𝐴𝐵= 140° and
m𝐶𝐷= 30°, then,
m∠𝐴𝑂𝐵=
1
2
(140°−30°)
m∠𝐴𝑂𝐵=
1
2
(110°)
m∠𝐴𝑂𝐵= 55°

9. Theorem 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 arcs.
m∠LMC= 12 (mLEC-mLG)
For example, if mLEC= 186°
and mLG= 70°, then,
m∠LMC= 12 (186°-70°)
m∠AOB= 12 (116°)
m∠AOB= 58°

10. Theorem 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.
m∠𝐾𝑄𝐻=
1
2
(m𝐻𝐽𝐾 −m𝐻𝐾)
For example, if m𝐻𝐽𝐾= 250° and
m𝐻𝐾= 110°, then,
m∠𝐾𝑄𝐻=
1
2
(250°−110°)
m∠𝐾𝑄𝐻=
1
2
(140°)
m∠𝑲𝑸𝑯= 70°

11. Theorem 4: if two secants intersect in the interior of a
circle, then the measure of an angle formed is ½ the sum
of the measures of the arcs intercepted by the angle and
its vertical angle.
If m𝑊𝑅= 100 and m𝑋𝑆= 120,
what is the measure of ∠1?
m∠1=
1
2
(m𝑊𝑅 + m𝑋𝑆)
m∠𝐴𝑂𝐵=
1
2
(100°+ 120°)
m∠𝑨𝑶𝑩= 110°

12. Theorem 5: if a secant and a tangent intersect at the
point of tangency, then the measure of each angle formed
is ½ the measure of its intercepted arc.
If m𝑄𝑆= 170, what is the measure of
∠QSR?
m∠𝑄𝑆𝑅=
1
2
(170)
m∠𝑸𝑺𝑹= 85°

13. “AREA OF A
SECTOR AND
SEGMENT OF A
CIRCLE”
^^

14. In the figure, what do you think is the
term given to the shaded part of the
circle?
The shaded part is called the sector.
To find the area of a sector of a circle,
get the product of the ratio
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑐
360
and the area of the circle.

15. For example, the radius of ⨀ C is 10 cm. If m𝐴𝐵=
60, what is the area of sector ACB?
a.
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑐
360
= 1/6
b. 𝐴 = 𝜋𝑟2
= 100 𝜋𝑐𝑚2
c. Area of sector ACB =(
1
6
) (100𝑐𝑚2
)
The area of sector ACB is
𝟓𝟎𝝅
𝟑
𝒄𝒎𝟐
.

16. A segment of a circle is a
region bounded by a chord
and the arc subtended by the
chord. It is a portion of the
area of a circle cut off by a
chord.
Look at the example.

17. EXAMPLE
^^

18. Step 1. Find the area of the sector.
Area of the sector =
𝑚
360
(𝜋𝑟2) =
60
360
(𝜋 · 52) =
25
6
𝜋 cm2
Step 2. Find the area of ∆ 𝑨𝑶𝑩.
A =
𝑠2
4
3 =
(5)2
4
3 =
25
4
3 𝑐𝑚2
Step 3. Find the area of the segment.
Area of the segment = Area of the sector – Area of ∆ 𝐴𝑂𝐵
=
25𝜋
6
-
25
4
3 = 2.264651843
(calculator value)
The area of the segment is approximately 2.3 𝐜𝐦𝟐.

19. THANK YOU ^^
~L~