Some properties of tangents, secants and chords, Angles formed by intersecting chords, tangent and chord and two secants, Chords and their arcs, Segments of chords secants and tangents, Lengths of arcs and areas of sectors

1. Some properties of tangents, secants and chords
A line in the plane of the
circle that intersects the
circle at exactly one
point is called tangent
line. The point of
intersection is called the
point of tangency.
The Tangent- Line Theorem
If a line is tangent to a circle, then it is perpendicular to the radius at its
outer endpoint. Or
If the tangent to a circle and the radius of the circle intersect they do so at
right angles :

2. Some properties of tangents, secants and chords
The Tangent- Line Theorem
If a line is tangent to a
circle, then it is
perpendicular to the radius
at its outer endpoint. Or
If the tangent to a circle and
the radius of the circle
intersect they do so at right
angles :

3. Some properties of tangents, secants and chords
Example 1.
Line AP is tangent to
circle C at P. If seg.
CP=7cm and seg.
AP=24cm, how far is
point A from a center C?

4. Some properties of tangents, secants and chords
Corollary:
The tangents to a circle at
the endpoints of a diameter
are parallel.
If line l and line m are perpendicular to a diameter
EP, then line l is parallel to line m.

5. Some properties of tangents, secants and chords
The Tangent-Segments
Theorem
1. The lengths of two
tangent segments from
an external point to a
circle are equal.
2. The angles between the
tangent segments and
the line joining the
external point to the
center of the circle are
congruent.
1. If line EF and line PF are
tangent to circle C at E and P
respectively intersects at F,
then EF = PF.

6. Some properties of tangents, secants and chords
The Tangent-Segments
Theorem
1. The lengths of two
tangent segments from
an external point to a
circle are equal.
2. The angles between the
tangent segments and
the line joining the
external point to the
center of the circle are
congruent.
2. If tangent segments EF and
PF form an angle EFP at F,
then angle EFC = angle CFP.

7. Angles formed by intersecting chords
a. m∠1 =
1
2
𝑎𝑟𝑐 𝐺𝐻 + 𝑎𝑟𝑐 𝐼𝑃
b. m∠2 =
1
2
𝑎𝑟𝑐 𝐺𝑃 + 𝑎𝑟𝑐 𝐻𝐼
If two chords intersect in
a circle, the angle they
form is half the sum of
the intercepted arcs.

8. Angles formed by intersecting chords
Example 2.
Find the value of x.

9. Angles formed by intersecting tangent and chord
m∠𝐻𝑃𝐾 =
1
2
𝑎𝑟𝑐 𝐻𝑃
Tangent Secant Theorem
If a chord intersects the
tangent at the point of
tangency, the angle it
forms is half the measure
of the intercepted arc.

10. Angles formed by intersecting tangent and chord
Tangent Secant Theorem
Example 3.
Find the value of x.

11. Angles formed by intersecting two secants
If two secants intersect
outside a circle half the
difference in the measures
of the intercepted arcs
gives the angle formed by
the two secants.
𝑚∠𝑞 =
1
2
𝑎𝑟𝑐 𝑃𝑄 − 𝑎𝑟𝑐 𝑂𝑅

12. Angles formed by intersecting two secants
Example 4
Find the value of x.

13. Example 5
Match the figure with the formula below.
b)
a)
d)
c)
1. 𝑚∠1 = 1/2(𝑚 − 𝑛)
2. 𝑚∠1 = 1/2(𝑛 − 𝑚)
3. 𝑚∠1 =
1
2
𝑚
4. 𝑚∠1 = 1/2(𝑚 + 𝑛)

14. Chords and their arcs
Theorem:
If in any circle two chords
are equal in length then
the measures of their
corresponding minor arcs
are same.
As shown in the figure, BC and DE
are congruent chords. Therefore,
according to the theorem stated
above m(arc BC) = m(arc DE) or
m∠BAC = m∠EAD.

15. Chords and their arcs
Theorem:
if two chords are
equidistant from the
center of the circle, they
are equal in measure.
In the figure, seg. PQ and seg. RS
are two chords equidistant
such that seg.OX = seg.OY
and perpendicular with center O,
therefore, according to the theorem
stated above seg. PQ = seg. RS

16. Chords and their arcs
Theorem:
The perpendicular from
the center of a circle to a
chord of the circle bisects
the chord.
In the figure, XY is the chord of a
circle with center O. Seg.OP is the
perpendicular from the center to the
chord. According to the theorem
given above seg XP = seg. YP.

17. Example 6
a. Find the degree
measure of each of the
five congruent arcs of a
circle around a regular
pentagon.
b. Solve the length of x.

18. Segments of chords, secants and tangents
The Intersecting
segments of Chords
Theorem
If two chords are
intersect, then the product
of the lengths of the
segments of one chord is
equal to the product of the
lengths of the segments of
the other chord.
𝑀𝑃 ∙ 𝑃𝑂 = 𝑁𝑃 ∙ 𝑃𝐿

19. Segments of chords, secants and tangents
Example 7
Solve the value of x.

20. Segments of chords, secants and tangents
The Segments of
Secants Theorem
If two secants intersect in
the exterior of the circle,
the product of the length
of one secant segment and
the length of its external
part is equal to the
product of the length of
the other secant and the
length of its external part. QU ∙ 𝑄𝑅 = 𝑄𝑇 ∙ 𝑄𝑆

21. Segments of chords, secants and tangents
Example 8
Solve the value of x.

22. Segments of chords, secants and tangents
The Tangent Secant
Segments Theorem
If a tangent segment and
a secant segment intersect
in the exterior of a circle,
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 part. 𝑃𝑇2 = 𝐵𝑃 ∙ 𝐴𝑃

23. Example 9
PT is a tangent
intersecting the secant
through AB at P.
Given
(seg. PA) = 2.5 cm.
and (seg.AB)=4.5 cm.,
find (seg PT).
Segments of chords, secants and tangents

24. Perimeter or circumference of a circle = 2𝜋𝑟 where r
is the radius or C=𝜋𝑑, where d is the diameter d=2r
Perimeter and Area of a Circle
Area of a circle = 𝜋r2 where ' r ' is the radius of the
circle
Area of a circle = 𝜋
𝑑
2
2 where ' d ' is the diameter of
the circle

25. Example 10
a. Find the circumference of a circle with area
25𝜋 sq. ft.
Perimeter and Area of a Circle
b. Find the area of a circle with circumference
30𝜋 cm.

26. An arc is a part of the
circumference of the
circle; a part
proportional to the
central angle.
If 3600 corresponds to
the full circumference.
i.e. 2 𝜋 r then for a
central angle of m0 the
corresponding arc length
will be l such that
Lengths of arcs and areas of sectors

27. Analogically consider the
area of a sector. This too
is proportional to the
central angle.
3600 corresponds to
area of the circle 𝜋 r2.
Therefore for a central
angle 𝑚0 the area of the
sector will be in the ratio
Lengths of arcs and areas of sectors

28. Example 11
In a circle with the
radius of 2 cm, the
central angle for an arc
AB is 750. Find (seg.AB).
Also find the area of the
sector AOB having a
central angle of 750
Lengths of arcs and areas of sectors