The circle third edition_025338.pdf

Baraka Loibanguti Mathematics teacher Page 1 of 20
Tanzania syllabus
Author: Baraka Loibanguti
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Email: barakaloibanguti@gmail.com
Mobile: +255714872887
Twitter: @bloibanguti
CIRCLES AND THEOREMS

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x
O
2x
A
B
2
a b
a O b
A
X
B
The circle theorems
Ordinary Level – Tanzania Secondary schools
At the end of this topic, learners should be able to: -
o To establish the following results and use them to prove further properties
andsolve problems:
o The angle subtended at the circumference is half the angle at the centre
subtended by the same arc
o Angles in the same segment of a circle are equal
o A tangent to a circle is perpendicular to the radius drawn from the point of
contact
o The two tangents drawn from an external point to a circle are the same
length
o The angle between a tangent and a chord drawn from the point of
contact isequal to any angle in the alternate segment
o A quadrilateral is cyclic (that is, the four vertices lie on a circle) if and only
if the sum of each pair of opposite angles is two right angles
o If AB and CD are two chords of a circle which cut at a point P (which
may beinside or outside a circle) then PA · PB = PC · PD
o If P is a point outside a circle and T, A, B are points on the circle such that
PT is a tangent and PAB is a secant then PT = PA · PB
These theorems and related results can be investigated
through a geometry package such as Cabri Geometry.
It is assumed in this chapter that the student is familiar
with basic properties of parallel linesand triangles
(a) ANGLE PROPERTIES OF THE CIRCLE P
Theorem 1
The angle at the centre of a circle is twice the
angle atthe circumference subtended by the
same arc.
Proof P
Join points P and O and extend the line through O as
shownin the diagram.
Note that AO = BO = PO = r the radius of the circle.
Therefore, triangles PAO and PBO are isosceles.
Let ∠APO = ∠PAO = a and ∠BPO = ∠PBO = b
Then angle AOX is 2a (exterior angle of a triangle) and angle
BOX is 2b◦
(exterior angle of a triangle)
∴ ∠AOB = 2a + 2b = 2(a + b) = 2∠APB
Note: In the proof presented above, the centre and point P are considered to be on
the same sideof chord AB.

Q
The proof is not dependent on this and the result always
holds.The converse of this result also holds:
i.e., if A and B are points on a circle with centre O and angle APB is equal to
half angle
AOB, then P lies on the circle.
• A segment of a circle is the part of the plane
bounded byan arc and its chord.
• Arc QPR and chord QR define a major
segment whichis shaded.
• Arc QTR and chord QR define a minor
segment which is not shaded.
∠QPR is said to be an angle in segment QPR.
Theorem 2
Angles in the same segment of a circle are equal.
Proof
Let ∠MNP = n and ∠MQP = q
Then by Theorem 1 ∠AOB = 2n = 2q
Therefore n = q
Theorem 3
The angle subtended by a diameter at the
circumference is equal to a right angle (90◦
).
Proof
The angle subtended at the centre is 180◦
.
Theorem 1 gives the result.
 XOY = 180O
,  XZY = 
2
1
XOY
 XZY = 
=

 90
180
2
1
Proved
P
X Y
Z
T
N q
n
M

A quadrilateral which can be inscribed in a circle is called a cyclic quadrilateral.
A
B
D
C ABCD is a cyclic quadrilateral
(four angles shape)

=
Theorem 4
The opposite angles of a quadrilateral inscribed in a circle sum to two right angles
(180). (Theopposite angles of a cyclic quadrilateral are supplementary). The
converse of this result also holds.
Proof
O is the centre of the circle
By Theorem 1:
y = 2b and x =2a
Also x + y = 360o
Therefore 2a + 2b = 360o
i.e. a + b = 180o
b
The converse states: if a quadrilateral has opposite angles supplementary then the
quadrilateralis inscribable in a circle.
Find the value of each of the pronumerals in the
diagram. O is thecentre of the circle and ∠AOB = 100◦
.
Solution
Theorem 1 gives that z = y = 50o
The value of x can be found by observing either of the
following.
z y
O
x B
Reflex angle AOB is 260o. Therefore x = 130o (Theorem 1)
or y + x = 180 (Theorem 4)
Therefore x = 180o − 50 = 130o
A, B, C, D are points on a circle. The diagonals of quadrilateral ABCD meet at X.
Prove thattriangles ADX and BCX are similar.
Solution
∠DAC and ∠DBC are in the same segment.
Therefore m = n
∠BDA and ∠BCA are in the same segment.
Therefore p = q
Also ∠AXD = ∠BXC (vertically opposite).
Therefore, triangles ADX and BCX are equiangularand
thus similar.
A
B
a
C
D
x
y
100o
n
m
q p
A

An isosceles triangle is inscribed in a circle. Find the angles in
the three minor segments of the circle cut off by the sides of
this triangle.
32
74 74
B C
74
Solution
First, to determine the magnitude of ∠AXC cyclic
quadrilateral AXCB is formed. Thus ∠AXC and X
∠ABC are supplementary.
Therefore ∠AXC = 106◦
. All angles in the minor
segment formed by AC will have this magnitude.
In a similar fashion it can be shown that the angles in the minor segment formed byAB
all have magnitude 106o
and for the minor segment formed by BC the angles all have
magnitude 148◦
.
A

50
y
x
z
y
O
108
x
z O
35
y
y
x
3x
y
z
x
59
112
y
x
x y
93
68
1. Find the values of the pronumerals for each of the following, where O denotes
the centre ofthe given circle.
a b c
d e f
2. Find the value of the pronumerals for each of the following.
a b c
PRACTICE EXERCISE 01
25
x
125

130 y
70
x

40
3. An isosceles triangle ABC is inscribed in a circle. What are the
angles in the three minor segments cut off by
the sides of thetriangle?
4.ABCDE is a pentagon inscribed in a circle. If AE = DE and ∠BDC = 20◦
,
∠CAD = 28◦
and ∠ABD = 70◦
, find all of the interior angles of the pentagon.
5. If two opposite sides of a cyclic quadrilateral are equal, prove that
the other two sides areparallel.
6. ABCD is a parallelogram. The circle through A, B and C cuts CD
(produced if necessary) at
E. Prove that AE = AD.
7. ABCD is a cyclic quadrilateral and O is the centre of the circle through A, B, C
and D. If
∠AOC = 120◦
, find the magnitude of ∠ADC.
8. Prove that if a parallelogram is inscribed in a circle, it must be a rectangle.
9. Prove that the bisectors of the four interior angles of a
quadrilateral form a cyclicquadrilateral.
A

T
S
A B
(b) LINES PROPERTIES OF A CIRCLE
Tangents
Line DC is called a secant and line segment AB a chord.
If the secant is rotated with D as the pivot point a
sequence of pairs of points on the circle is
defined. As DC moves towards the edge of the
circle the points of thepairs become closer until
they eventually coincide.
When PQ is in this final position (i.e., where
the intersection points A and B collide)
it is called a tangent to the circle.
Line D touches the circle. The point at which the tangent
touches the circle is called the point of
contact. The length of a tangent from a point
P outside the tangent is the distance between
P and the point of contact.
Theorem 5
A tangent to a circle is perpendicular to the radius drawn to the point of contact.
Proof
Let T be the point of contact of tangent PQ.
Let S be the point on PQ, not T,
such that OSP is a right angle. P
Triangle OST has a right angle at S.
Therefore OT > OS as OT is the hypotenuse of triangle OTS.
∴ S is inside the circle as OT is a radius.
∴ The line through T and S must cut the circle again. But PQ is a tangent. A contradiction.
Therefore T = S and angle OTP is a right angle.
Theorem 6
The two tangents drawn from an external point to a circle are of the same length.
Proof
Triangle XPO is congruent to triangle XQO as
XO isa common side. ∠XPO = ∠XQO = 90O
OP = OQ (radii)
Therefore, XP = XQ (third side of the triangle)
D C
B1
B2
B3
B4
X
B5
A1
A2
A3
A4
A5
D

Alternate segment theorem
The shaded segment is called the alternate
segment inrelation to ∠STQ.
The unshaded segment is alternate to ∠PTS
Theorem 7
The angle between a tangent and a chord drawn
from the point of contact is equal to any angle in the
alternate segment.
Proof
Let ∠STQ = x◦
,∠RTS = y◦
and ∠TRS = z◦
where RT is
a diameter.
Then ∠RST = 90◦
(Theorem 3, angle subtended by a
diameter)
Also ∠RTQ = 90◦
(Theorem 5, tangent is perpendicular to
radius)
R
X
z
O S
Hence x + y = 90 and y + z
= 90
Therefore x = z
P T Q
But ∠TXS is in the same segment as ∠TRS and therefore ∠TXS = x
Solution
Triangle PTS is isosceles (Theorem 6, two tangents from the same point) and
therefore ∠PTS = ∠PST
Hence y = 75.
The alternate segment theorem gives that x = y = 75
Find the values of x and y.
PT is tangent to the circle centre O
Solution
x = 30 as the angle at the circumference is
halfthe angle subtended at the centre and y = 60 as
∠OTP is a right angle.
T Q
P
y
x
Example 5
T
O
x
y
60o
P
Find the magnitude of the angles x and y in
the diagram.
Q x 30
y

x y
81 73
w
80
x
40
y z
The tangents to a circle at F and G meet at H. If a chord FK is
drawn parallel to HG, prove that triangle FGK is isosceles.
Solution
Let ∠XGK = y
Then ∠GFK = y (alternate segment theorem)
and ∠GKF = y (alternate angles).
Therefore, triangle FGK is isosceles with FG = KG
1 Find the value of the pronumerals for each of the following. T is the point of
contact of thetangent and O the centre of the circle.
a b
d e
Q
c
Example 6
K
F
H
G X
Y
y
S and T are points
of contact of
tangentsfrom P. TP
is parallel to QS
T
O
x
33
q
T
T
BC = BT
P
2 If AB and AC are two tangents to a circle and ∠BAC = 116◦
, find
the magnitudes of the angles in the two segments into which BC
divides the circle.
3 From a point A outside a circle, a secant ABC is drawn cutting the circle at B
and C, and atangent AD touching it at D.A chord DE is drawn equal in length
to chord DB. Prove thattriangles ABD and CDE are similar.
4 AB is a chord of a circle and CT, the tangent at C, is parallel to AB. Prove that
CA = CB.
z
y
74 x
w
z
54
x
y

5 A triangle ABC is inscribed in a circle, and the tangent
at C to the circle is parallel to the bisector of angle
ABC.
a) Find the size of ∠BCX.
b) Find the magnitude of ∠CBE, where E is the
point of intersection of the bisector of angle ABC
with AC.
c) Find the magnitude of ∠ABC.
6 AT is a tangent at A and TBC is a secant to the circle.
Given ∠CTA = 350,∠CAT = 1150, find the magnitude
of angles ACB, ABC and BAT.
B X
A 40
C
7 Through a point T ,a tangent TA and a secant TPQ are drawn to a circle
AQP. If the chord
AB is drawn parallel to PQ, prove that the triangles PAT and BAQ are
similar.
8 PQ is a diameter of a circle and AB is a perpendicular chord cutting
it at N. Prove that PN isequal in length to the perpendicular from
P on to the tangent at A.
CHORDS IN CIRCLES
Theorem 8
If AB and CD are two chords which cut at a point P (which may be
inside or outside the circle)then PA × PB = PC × PD.
Proof
CASE 1 (The intersection point is inside the circle.)
Consider triangles APC and BPD.
∠APC = ∠BPD (vertically opposite)
∠CDB = ∠CAB (angles in the same segment)
∠ACD =∠DBA (angles in the same segment)
Therefore, triangle CAP is similar to triangle BDP.Therefore
PB
CP
PD
AP
= and AP×PB = CP×PD,
which can be written PA×PB = PC×PD
E
C
A T
C
A
D
P
B

2
CASE 2 (The intersection point is outside
the circle.)Show triangle APD is similar to
triangle CPB
Hence
A
PB
PD
CP
AP
= i.e. AP ×PB = PD×CP
which can be written PA ×PB = PC ×PD
Theorem 9
If P is a point outside a circle and T, A, B are points on the circle
such that PT is a tangent and PAB is a secant then PT2
= PA ×PB
Proof
∠PTA = ∠TBA (alternate segment theorem)
∠PTB = ∠TAP (angle sum of a triangle) P
Therefore, triangle PTB is similar to triangle PAT
PT
PA
=
PB
PT
which implies PT = PA · PB
The arch of a bridge is to be in the form of an arc of a circle. The span of the
bridge is to be30m and the height in the middle 4 m. Find the radius of the
circle.
Solution
Theorem 8
AP × PQ = MN × PN
4 × PQ = 15 × 15
PQ = 25
.
56
4
225
=
PQ = 2r – 4 = 56.25
Therefore, the radius, r = 30.125 m
Example 7
Q
A
P
15 m
4 m
15 m
∴

of 20
The Circle third edition 2021 Baraka Loibanguti
THE CIRCLE
If r is the radius of a circle, with center O, and if A is any
point inside the circle, show that theproduct CA ·AD = r2 −
OA2, where CD is a chord through A.
Solution
Let PQ be a diameter through A
Theorem 8 gives that
CA · AD = QA · AP
Also, QA = r − OA and PA = r + OA
∴ CA ·AD = r2 − OA2
1 If AB is a chord and P is a point on AB such that AP = 8 cm, PB = 5
cm and P is 3 cmfrom the centre of the circle, find the radius.
2 If AB is a chord of a circle with centre O and P is a point on AB such
that BP = 4PA,
OP = 5 cm and the radius of the circle is 7 cm, find AB.
3 Two circles intersect at A and B and, from any point P on AB
produced tangents PQ and PR are drawn to the circles. Prove that
PQ = PR.
4 PQ is a variable chord of the smaller of two fixed concentric circles.
PQ produced meets the circumference of the larger circle at R. Prove
that the product
RP × RQ is constant for all positions and lengths of PQ.
5 Two chords AB and CD intersect at a point P within a circle.
Given that: -
a) AP = 5 cm, PB = 4 cm, CP = 2 cm, find PD
b) AP = 4 cm, CP = 3 cm, PD = 8 cm, find PB.
6 ABC is an isosceles triangle with AB = AC. A line
through A meets BC at D and thecircumcircle of the
triangle at E. Prove that AB2 = AD × AE.
D

of 20
THE CIRCLE
WORKED EXAMPLES
1. If O is the center of the circle, Calculate the size of angle DAB
2. A circle of diameter 10 cm has a chord drawn inside it. The chord is 7 cm long.
a) Make a sketch to show this information.
b) Calculate the distance from the midpoint of the chord to the centre
of the circle.
3. The diagram shows the circle with center O. PT and RT are tangents to the
circle, angle ROP = 144O.
Work out the size of the angle PRT marked x.
R
P
T
O 144O
x
D
A O
B
C Solution
Arc DC = 2 × angle DBC
Arc DC = 30o
Arc AD = arc CB (parallel chords subtends equal arcs)
Arc AD + arc DC + arc CB = 180o (semi-circle)
2arc AD + arc DC =180o
2arc AD + 30o = 180o
Arc AD = 75o
Angle DAB is subtended by minor arc DB
Arc DB = arc DC + arc CB
Minor arc DB = 75o + 30o = 105o
Angle DAB =
2
1
× 105o = 52.5o
150
10 cm
7 cm
Construct OD and OE where OD is the
radius of the circle and angle OED = 90o,
therefore, ΔOED is a right-angled triangle. E
bisect the chord CD, thus EC = ED = 3.5cm.
By Pythagoras theorem:
OD2 = OE2 + ED2
52 = OE2 + 3.52
OE2 = 12.75cm, thus OE = 3.6cm
A B
C D
O
E
OP = OR (radii)
PT = TR (tangents from
the external point of a circle)
Angle OPT = 90o
Angle ORT = 90o
144o + 90o + 90o + x =360o
(quadrilateral)
Thus, x = 36o

of 20
THE CIRCLE
4. PT and RT are the tangents to the circle with center O. Angle PTR 32O, find
the size of angle labelled y
5. Points A, B and C are on the circle and O is the center of the circle. Angle BCE
= 63o. FE is a tangent to the circle at point C.
a) Calculate the size of the angle ACB. (Give reasons)
b) Calculate the size of the angle BAC (Give reasons).
6. P, Q, R and S are points on the circumference of a circle; centre O. PR is a
diameter of the circle. Angle PSQ = 56O
S
R
P
T
O
y 32o
Construct OP and OR
Angles OPT + POR +
ORT + RTP = 360o
Angle OPT = ORT = 90o
(tangent radius)
180o + 32o POR = 360O
POR = 148O
Angle PSR =
2
1
angle POR
(theorem)
PSR = y = 74o
O
A B
C
F
E
65O
(a) Angle ACB + BCE = 90o
(tangent radius)
ACB + 65o = 90o
Angle ACB = 25o
(b) Angle BAC = BCE (alternate
segment)
Thus, BAC = 65o
56o
S
P
Q
R
O
(a) PQR = 90o (angle at a semi-circle)
(b) PRQ = 56o (same segment as PSQ)
(c) POQ = 2×56o = 112o (Central-circumference
angle)

of 20
THE CIRCLE
a) Find the size of angle PQR. (Give a reason for your answer)
b) Find the size of angle PRQ. (Give a reason for your answer)
c) Find the size of angle POQ. (Give a reason for your answer)
7. A, B, C and D are four points on the circumference of a circle.
ABE and DCE are straight lines. Angle BAC=25°. Angle EBC=60° and
angle CBD=65°.
a) Find the size of angle ADC.
b) Find the size of angle ADB
(c) Vanessa says that BD is a diameter of the circle.
Is Vanessa correct? Explain your answer!!
8. The diagram shows a circle, centre O. AC is a diameter. Angle BAC=35°. D is the
point on AC such that angle BDA is a right angle.
a) Work out the size of angle BCA. Give reasons for your answer.
b) Calculate the size of angle DBC.
c) Calculate the size of angle BOA
60o
25o
A
B
C
D E
Solution
Angle ABD = 180o – 60o -65o = 55o
(straight line)
Angle BCD = 180o – 65o – 25o = 90o
(triangle)
Minor arc AB = 2 × 35o = 70o
Minor arc BC = 2 × 25o = 50o
Minor arc AC = Arc AB + Arc BC
(a) ADC =
2
1
× (70o + 50o) = 60o
(b) Angle ADB = 60o – 25o = 35o
(c) Vanessa is correct, because the
angle at semi-circle is 90o, thus
angle DCB = 90o
A
B
C
D
35o
O
Solution
(a) ABC = 90o (semi-circle)
35o + 90o + BCA = 180o
thus, angle BCA = 55o
(b) DBC + BCA + CDB = 180o
DBC + 55o + 90o = 180o
DBC = 35o
(c) Angle BOA = 90o

of 20
THE CIRCLE
9. A, B, C and D are four points on the circumference of a circle. TA is the
tangent to the circle at A. Angle DAT=30°. Angle ADC=132°.
a) Calculate the size of angle ABC. Explain your method.
b) Calculate the size of angle CBD. Explain your method.
c) Explain why AC cannot be a diameter of the circle
10. Points A, B and C lie on the circumference of a circle with centre O. DA is
the tangent to the circle at A. BCD is a straight line. OC and AB intersect at
E.
Angle BOC=80°. Angle CAD=38°.
(a) Calculate the size of angle BAC.
(b) Calculate the size of angle OBA.
(c) Give a reason why it is not possible to draw a circle with diameter ED through the
point A
30o
132o
A
B
C
D
T
Solution
(a) Angle ABC = 180o – 132o = 48o
(opp. Angles of cyclic
quadrilateral).
(c) AC cannot be a diameter because
- The semi-circle angle CDA is
not 90o, it is 132o
- The other semi-circle angle
CBA = 48o so AC cannot be a
diameter.
Answers:
(a) BAC = 40O
(b) OBA = 12O
B
A
C
D
O

of 20
THE CIRCLE
11. Calculate the area of the minor segment AB if chord AB is 8 cm
O
50o
C
A B
Solution
Construct AO and BO (radii)
The central angle AOB
Find the area of the triangle AOB
Find the area of the sector AOB
Area of minor segment = Area of
sector – area of triangle
Answer: 10.3 square centimeter
OTHER MATHEMATICS TOPIC NOTES AVAILABLE
FOR O-LEVEL
1) All form 1 Mathematics + ICT topics
2) All form 2 mathematics + ICT topics
5) All additional mathematics topics
FOR A-LEVEL
1) All form 5 advance mathematics topics
2) All form 5 BAM topics
3) All form 6 advance mathematics topics
4) All form 6 BAM topics
PHYSICS NOTES AVAILABLE
FOR O-LEVEL
1) Physics form 1 full notes
PHYSICS + ICT FOR A-LEVEL
*Preparation in progress until January 2022
Author: Baraka Loibanguti
For more contact me:
Email: barakaloibanguti@gmail.com
Mobile: +255714872887
Twitter: @bloibanguti

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The circle third edition_025338.pdf