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CIRCLE
DEFINITION AND PROPERTIES OF A CIRCLE
A circle can be defined in two ways.
A circle: Is a closed path curve all points of which are equal-distance from a fixed point called centre OR
– Is a locus at a point which moves in a plane so that it is always of constant distance from a fixed point known as a centre.
O – Is called the centre of the circle.
OP – Is the line segment is the radius.
AB – Is the line segment of diameters of the circle.
O – (Centre) – Is a fixed point of circle.
OP (Radius) – Is the constant distance from the centre to any point on a circumstance of a circle.
AB (Diameter) – Is a line segment which passes through the centre of a circle.
A circumference – Is a length of a locus which moves around the centre.
Diameter =2 x Radius
D = 2r
Hence the diameter of a circle is equal to two times radius.
RS (Secant) – Is a line segment whose points are on the circle.
BOC – Is called a central angle
(PTQ) – A Segment – Is the part of a circular region included within the chord and its arc.
(COB) Sector – Is the part of a circular region bounded by two radii and an arc.
CENTRAL ANGLE
Consider a circle of radius r, length of arc l, subtending a central angle.
– The length of the circumference C of the circle is C = 2πr. This means that the length of the arc intercept by a central angle 360º is 2πr.
– The length of an arc is proportional to the measure of the central angle of the central angle. Thus if the central angle is 1
Example 1.
An arc subtends an angle of 200 at the centre of a circle of radius 25m.
Find the length of the arc.
Length of an arc is given by
= Length of an arc = 8.72m
An arc of length 5cm subtends 50º at the centre of a circle. What is the radius of the circle.
Data
θ = 50º
Length = 5 cm
Radius = required
Length of an arc is equal to
=5.73cm
Example 3
A circular running track has radius 50m. A sprinter runs 100m along the track.
Through what angle has she turned?
Radius = 50m
Length of an arc = 100m
θ = Required
Length at an arc is equal to
The central angle is 114 .650
Questions:
1. An arc subs tends 25 at the centre of a circle of radius 40m. What is the length of the arc?
Data
Length of an arc = required
Radius = 40m
θ =- 25
Length at an arc is equal to
The length at an arc = 17.44m
The length at an arc = 17.44m
2. An arc of length 17cm forms a circle of radius 40cm what angle does the arc subtend?
Data :-
Length of an arc = 17cm
Radius = 40cm
θ= Required
Length of arc is equal to
The arc subtends 24.330
3. An arc of length 16m subtends 400 at the centre of the circle. What is the radius at the circle?
Data
Length an arc = 16m
Radius = Required
θ = 400
Length of arc is equal to
The radius = 22.85m
The radius = 22.85m
RADIAN MEASURE
Angles can also be used to measure the amount of turning. Turns of a minute hand of a clock and a wheel can be measured in both angles and radians. Example a minute hand of clock turns through an angle of 90º 0f ½ π radians between noon and 12:15pm as shown in the figure.
– The tip of the hand has covered a distance of ½ π radians.
From noon to 12:45 pm, the hrs turned through an angle of 270ºor 3/2πr radians. The angle 270º is reflex angle.
– One complete turn at hand clock represents an angle of 360º or 2π radians.
– Measures of angles more than 3600 or 2 π radius can be obtained if hand of a clock measures more than one complete turn. Example from noon to 1:15 pm, the hand has turned through 1 ¼ turns. Now one turn is 3600 or 20 radians.
– 5/4 turns or 3600 x 5/4 0r 2π x 5/4 radians which reduce to 450º 0r 5/2π radians.
– There fore from noon to 1:15 pm the hand turns through 450º or 5/2π radians.
Questions
1. Give the size in degree at an angle through which a minute hand of a clock has returned between noon and the following times.
(a) 12:40
Solution:
1min = 6
40min =?
X= 40min x 6
X = 240
(b) 3:00
Solution
1hour = 360
3 hour =?
X = 3 hour x 3600
X = 10800
(c) 9:00
Solution
1hour = 3600
9 hour =? X
X = 9 x 3600
X = 32400
2. Give the size in radians at angles through which the minute hand of a clock has turned between noon and the following times.
(a) 12:20pm
Solution
1min = 60
20 min =? X
X= 20min x 6
X = 1200
πrad = 180º
? = 120º
x = π rad x 120
X = 2/3 π rad
(b) 2:15
Solution
1hour = 60 min
2 hour =? X
x =60 x 15min
X =120
120 x 6º=720º
= 7200 +90º
= 810º
120 x 6º=720º
= 7200 +90º
= 810º
(c) 24:00 noon
Solution
1 hour = 3600
24hrs = ?
X = 3600 x 20
= 86400
πrad = 1800
?x = 86400
x = 86400 x πrad
X = 48πrad
RADIAN MEASURE
– The relation between the arc length l, the central angle θ, and the radius r, can be used to compare the measurement of an angle in radius with the measurements in degree.
Circumference of the circle for the given radian C = 2πr. Circumference sector
But C = length at an arc
In abbreviation is written as:-
S = is radian measured of an angle
θ = is the degree measure
θ= 17 .190
θ= 17 .190
The angle in degree is 17. 190
Class Activity:-
1.Find the degree of each of the following :-
(i) 3/2π
Solution:-
= 900 x 3
= 3/2π = 2700
(ii) ¾π
(ii) ¾π
Solution
¾ π = 1350
Example 1:
Find in radian as multiple of π for each of the following degrees.
(a) 3150 (b) 2400
Solution:
(b) 2400
Solution:-
2. Change the following radians into degree
(a) 0.3 (b) 5
solution
Class activity.
1. Find the degree of each of the following
i) 3/2π
Solution.
ii 3/4 π
Solution
iii) 2 π
Solution
2.Find the radians multiple of the π following
(i)80°
Solution
Solution
ii) 215º
Solution
iii) 600
Solution
3. (i) Change the radians into the degree of 0.3
∴The angle the degree = 540
(ii) 5
Solution
Solution
The angle degree = 9000
ANGLES IN CYCLIC QUADRILATERAL
These are four angles whose vertices are lying on the circumference of a circle.
The angles p, q, r, and t are called cyclic angles in a quadrilateral ABCD, q and t, p and r are opposite angles.
1. THEOREM:
The opposite angles of a cyclic quadrilateral are supplementary (add up to 1800).
Given; A quadrilateral SPQR inscribed in a circle centered at 0
Required to prove: x+ y = 1800
Constuction; join OR and OP
Proof: in the above figure
a = Zx (angle s on a circle) PQR)
b = Zy (angle on arc PSR)
a+b = Zx + 2y but a+ b = 360º
3600 = 2 (x+y ) divide by 2 both sides
x+y = 1800
∴ x+ y = 180º hence proved.
Example:
Find the size of each lettered angle.
The opposite angles A and C, B and D
:. x +830 = 1800
y = 1800 – 83
y = 970
x+1070 = 1800
x= 1800 – 19
x = 630
2. THEOREM:
Any inscribed angle in a semicircle is a right angle.
Given
AB – Is a diameter at a circle.
O – Is the center.
C – Is any point on the Circumference
Required to prove that ACB = 900
2 ACB= AÔB (Angle at the centre)
But,
AÔB = 180o (Straight angle)
ACB = 90o Hence proved.
Example:
Calculate the Value of x
Soln:
PQR = 90o (Angle in a semi Circle)
Then 90o + 30o + x = 180o
(Angles sum in A)
x = 180o – 120o
x = 60o
An exterior angle is an angle formed outside a cyclic quadrilateral. An internal angle is formed inside the cyclic quadrilateral when you provide line from this angle you will form an angle which is outside called exterior angle.
THEOREM: 3
Exterior angle of a cyclic quadrilateral is equal to the inside opposite angle.
Aim: to prove that PSR = PQT
Proof: let angle PQR = a
Angle PRS = b
Angle PQT = y
(i) PQR + PSR = 180o (Because it is opposite angle of cyclic quadrilateral).
Thus a+b = 180o
(ii) Therefore when you equate them since both are 180o
(iii) PQR + RQT = 180o (Because they are adjacent angles on straight line)
Thus a + y = 180o
Therefore when you equate them since both are 180o you will have:
a + b = a + y
B = y
But b = PSR
And y = PSR
PSR = PQT
Proved.
THEOREM 4:Angles in the same segment are equal.
THEOREM 5: Angles in the semi circle are right angled triangle.
Aim: To prove that PxQ = PyQ
Construction: join OP and OQ
Let PÔQ = P
PVQ = q
PYR = r
(i) PÔQ = 2p x Q and
P = 2q (Because angle at the centre is twice the angle at the circumference)
(ii) PÔQ = 2PyQ (Because angle at the centre is twice the angle at the circumference)
P = 2r
x = P = 2q, P = 2r
= 2q = 2r
q = r
TO ANSWER THE QUESTIONS:
1. Find the value of X. if O is the centre of Circle
Soln:
Angle at the centre is twice the angle at the circumference.
Since the circle at the centre is 360o
3 x + 2x = 360o
5x = 360o
= 72o
(b) Angle at the centre is twice the angle of the circumference.
Opposite angle at a cyclic quadrilateral are supplementary (add up to 180o)
x + 3x = 180o
4x = 180
x=45
Soln:
Angle at the same segment are equal.
Angle at the centre is twice the angle at the circumference.
2 x + 260o = 360o
Since the circle at the centre is 360o
2x + 260o = 360o
2x = 3600-2600
2x=1000
x = 50o
Soln:
Angle at the centre is twice the angle at the circumference.
150o x 2 = 300o
Since the circle at the centre is 360o
x + 300o = 360o
x = 360o – 300
x = 60o
Soln:
Angle at the centre is twice the angle at the circumference.
40o x 2 = 80o
Since the circle at the centre is 360o
x + 80o = 360o
x = 360o – 80o
x = 280o
(f) Find the value of angles marked x, y and z
Soln:
Angle at the centre is twice the angle at the circumference.
X x 2 = 224o
X = 112o
Since the circle at the centre is 360o
y + 224o = 360o
y = 360o – 224o
y = 136oOpposite angle of a cyclic quadrilateral are supplementary (add up to 180o)
Z + 112o = 180o
Z = 180o – 112
z = 68o(g) Find ‘y’
Soln:
Angle at the centre is twice the angle at the circumference.
y x 2 = 2y
Since the circle at the centre is 360o
4y + 2y = 360o
y = 60oClass activity
1. MN is a diameter of a circle and L is a point on the circle. If MNL = 135o,
Find NML
Soln:
Any inscribed angle in a semicircle is a right angled triangle.
let NML=X
let NML=X
X + 35o + 90o = 180o
X + 125o = 180o
X = 180o – 125o
NML = 55o2. AB is a diameter of a circle radius 10 cm and e is a point on the circumference.
CB = 12, find CA (Remember Pythagoras theorem)
Soln:
By using Pythagoras theorem
202 = CA2+ 122
400 = CA2+ 144
CA2= 400 – 144
CA2 = 256
CA= 16cm3. AB is a diameter of a circle radius 40cm and C is point on the circumference.
If CBA = 62O, then find CÂB
Soln:
Let X = CÂB
X + 62o + 90o = 180o
X + 152o = 180o
X = 180o – 152o
CÂB = 28oTHE CHORD PROPERTIES OF A CIRCLE
The chord of a circle is the line segment whose end point are on the circle. A chord which passes through the centre of a circle is called a diameter. It is very important for you to know what a chord is and how to identify the chord properties of a circle because it will summarize you with this unit.
Therefore in this section you are going to study about the chord itself and the chord properties of a circle.
You are also going to study how to develop theorem which relate to these properties at chord. At the end of the section you will be able to identify the chord, prove the theorem of the chord. Properties in a circle and then apply these theorems on solving related problems in order to identify the properties of the chord properties it easier if you draw a circle with centre O.
You can see that O is a centre of the OM is the radius of the circle and PQ is chord of the circle.
Therefore you will discover that.
(a) The centre of the circle lies on the perpendicular bisector of the chord.
(b) The perpendicular from the centre of the circle to the chord
(c) The line joining the centre of the circle to the midpoint of the chord.
Then from the information above you can develop the theorem which can be written as;
THEOREM
The perpendicular bisector of a chord passes through the centre of the circle.
AIM: To prove that OMP = OMQ = 90o
Construction: Join OP, OQ and OM
Proof:
(i) OP = OQ (Radii)
(ii) PN = QM (M is midpoint given)
(iii) ON = OM (common)
(iv) OPM = OQM (Bisected angles)
Construction: Join OP, OQ and OM
Proof:
(i) OP = OQ (Radii)
(ii) PN = QM (M is midpoint given)
(iii) ON = OM (common)
(iv) OPM = OQM (Bisected angles)
The corresponding angle are congruent and hence 
OMP = OMQ = 90o proved.THEOREM: Parallel chords intercept congruence arc.
Aim: To prove that arc PR≡Are
Proof:
Arc AQ≡ Arc AP (AOB is diameter)
Arc AS≡ Arc AR (AOB is a diameter)
Arc PR ≡Arc AR – Arc AP and also
Arc QS≡ Arc AS – Arc AQ
By step (i) up to step (iii) above you can conclude that Arc PR≡ Arc QS proved
Class Activity:
Two chords, AB and CD of the circle whose radius is 13cm are equal and parallel.
If each is 12cm long, find the distance between them.
Two chords, AB and CD of the circle whose radius is 13cm are equal and parallel.
If each is 12cm long, find the distance between them.
Soln: – By using Pythagoras theorem
AD2= AB2 +DB2
262 =122 + DB2
676 = 144 + DB2
DB2= 676 – 144
DB2 = 532
Square root both sides
2.A chord of length 32cm is at a distance of 12cm from the centre of a circle.
Find the radius of a circle.
AD2= AB2 +DB2
262 =122 + DB2
676 = 144 + DB2
DB2= 676 – 144
DB2 = 532
Square root both sides
2.A chord of length 32cm is at a distance of 12cm from the centre of a circle.
Find the radius of a circle.
Soln:
By using Pythagoras theorem
AO2 = 122+ 162
= 144 + 256
= 400
Apply square root both sides
3. A distance of a chord PQ from the centre of a circle is 5cm.
If the radius of the circle is 13cm. Find the length of PQ
AO2 = 122+ 162
= 144 + 256
= 400
Apply square root both sides
3. A distance of a chord PQ from the centre of a circle is 5cm.
If the radius of the circle is 13cm. Find the length of PQ
Soln:
By using Pythagoras theorem.
132 = 52 + PQ2
169 = 25 + PQ2
PQ2= 169 – 25
PQ2= 144
Apply square root both sides
PQ = 12cm
Since OS is perpendicular to PQ
Therefore PS = SQ
By using Pythagoras theorem.
132 = 52 + PQ2
169 = 25 + PQ2
PQ2= 169 – 25
PQ2= 144
Apply square root both sides
PQ = 12cm
Since OS is perpendicular to PQ
Therefore PS = SQ
PQ = 12cm + 12cm= 24cm.
4. The chord AB of a circle with centre O radius 3cm long.
Find the distance of AB from O. give your answer in cm form
Soln:
By using Pythagoras theorem
a 2 +b2 =c2
15+b2 =9 x 3
b2 =27-15
Apply squar
e root both sides
e root both sides
∴ 
5.Two chords AB and CD of a circle with centre O. if AB = 10cm,
CD = 6cm, AO = 7cm. Find the distance between two chords
Triangle (i)
By Pythagoras theorem
a2 x 6 = C2
52 + b2 = 72
25 + b2 = 49
b2 = 49 – 25
b2 = 24
triangle (ii) by Pythagoras theorem
a2 + b2 = c2
32 + a2 = 72
9 +a2 = 49
a2 = 49 – 9
a2 = 40
Class Activity:
XY and PQ are parallel chords in a circle of centre O and radius 5cm.
If XY = 8cm and PQ = 4cm, find the distance between two chords.
Soln:
1st triangle
By using Pythagoras theorem.
C2 = a2 + b2
52 = 42 + b2
25 = 16 + b2
b2 = 25 – 16
b2 = 9cm
square root both sides
b= 3cm
2nd triangle
By using Pythagoras theorem.
C2 = a2 + b2
52 = 22 + b2
25 = 4 + b2
b2 = 25 – 4
Example:
1. AB is a chord of circle with centre X. the midpoint of AB is m.
If XAB find MXA
XAM + MXA = 180O
52o + 90o + MXA = 180o
MXA = 180o – 142º
= 38o
2. AB is a chord in a circle with centre C. the length of AB is 8cm and the
Radius of the circle is 5cm. find,
The shortest distance of AB from C
ACB
By applying Pythagoras theorem.
42 +mc2 = 52
Mc2 = 25 – 16
Mc2= 16
Square root both sides
Mc = 4m
Mc2 = 25 – 16
Mc2= 16
Square root both sides
Mc = 4m
Sin ACM = 0.8000
ACM = Sin-1 ((0.8)
= 53o X 2
= 106o
A chord AB has length 12m. it is 7m from the centre of the circle.
Find the (a) length of AC
(b)ACB
Soln:
By using Pythagoras theorem.
Using:
Tan ACM = 0.8571
ACM = Tan-1
= 40o
ACB =400 x 2= 80o
EXERCISE
1. M is the Centre at the chord AB at a Circle with centre X if
2. M is the centre at chord PQ at a circle with Centre O. if < PQO = 43O. Find
3. A circle has radius 13cm and centre X. a chord AB has length 24cm. find:-
(a) T distance of the chord from the Centre
(b) < AXB
By Pythagoras theorem
(AM)2 + (MX)2 = (AX)2
122 + (MX)2 = 132
144 + (MX)2 = 169
(MX)= 169 – 144
(MX)2 = 25
Square root both sides
(MX)2 = 25
MX = 5cm
The distance from the centre to chord is 5cm.
QUESTIONS:
1. Let A be the Centre at a chord PQ at a circle with Centre O. If < PQO = 43o, find < POQ
Soln
1. Q is the centre at a Circle and AB is a Chord
(a) The length at AB
(b) The distance at A from C
Soln:
3. PQ is a Chord in a Circle with centre R. PQ = 14cm and the distance at R from PQ is 9cm, find:-
(a) The radius at a circle
(b) 
Soln:
Let RQ = Radius
By using Pythagoras theorem
Using, SO TO CA
H A H
H A H
TANGENT PROPERTIES
A tangent to a circle touches it at exactly one point
THEOREM:
A tangent to a circle the line perpendicular to the radius at the point of contact.
TAP is a line perpendicular to the radius CA show that TAP is a tangent as follows:-
If b is another point on TAP then CB is the hypotenuse at A CAB and hence CB is longer than CA it follows that B lies outside the Circle. Hence TAP needs the Circle only at A. TAP is a tangent to the Circle.
Hence a tangent is perpendicular to the radius.
Examples:
1. TA is a tangent to the Circle with centre C. If 
Line AT⊥AC = 90°
90° + 49° +
90° + 49° +
139° + =180°
∴ = 41°
∴
2. A point T is 8cm from the centre C of a circle of radius 5cm. Find
(a) The length of the tangent from T to the circle
(b) The angle between the tangent and TC
By using Pythagoras theorem
(AC)2 + (AT)2 = (TC)2
52 + (AT)2 = 82
25 + (AT)2 = 64
AT2 = 64 – 25
(AT)2 = 39
AT = 6.24
=39º
=39º
QUESTIONS:
1. TA is a tangent to the circle at A. the centre is C. if
(a) Find 
0 0 + 32º =1800 0– 1220
0 
(b) If TC = 8cm, Find AT and radius of the circle
Solution: (a)
Line AT ⊥ AC = 900
Solution: (b)
= 8 (0.5299)
AT = 4.2392cm
2.TA is a tangent to a circle at A. The centre at the circle C if angle ACT = 73°and radius of the circle is 2m. find:-
(a) 
(b) TA and TC
Line TA 
AC = 900
90° + 73° + 0 = 1800 0 – 1630
AC = 90090° + 73° +
Solution: (b)
(i)
(i)
Opp = 2 Tan 730
=2(3.2709)
TA = 6.5418m
(ii) By using Pythagoras theorem
TA = 6.5418m
≈7m
C2 = a2 + b2
= 22 + 72
= 4+49
c2= 53
Square root both sides
C2 = 53
3. A point T is 6m form
the center C of a circle radius 3cm. Find:-
the center C of a circle radius 3cm. Find:-
(a)The length of Tangent from T to the circle
(b)The angle between the tangent and TC
Solution: (a)
Using Pythagoras theorem
A2 + b2 =C2
32 + b2 = 62
9 + b2 = 36
b2 = 36 – 9
b2 = 27
Square root both sides
b2 = 27
Solution: (b)
Cos = 0.5
= Cos -1 (0.5)
= 300
Angel between tangent and TC is 300
Class Activity
1.A point, 10m from the center X of circle at radius 6m. A tangent is drawn from P to the circle touching at A. Find the length of the tangent from P to the circle.
By using Pythagoras theorem
C2 = a2 + b2
102 = 62 + b2
100 = 36 + b2
b2 = 100 – 36
b2 = 64
Square root both sides
b2 = 64
b = 8m
∴The length of the tangent from P to the circle is 8m
∴The length of the tangent from P to the circle is 8m
2.A tangent is drawn from T to a circle of radius 8cm. The length of the tangent is 4cm. Find,
(a) The distance of T from the Centre C of the circle
(b) The angle between TC and the tangent
(a) By using Pythagoras theorem
C2 = a2 + b2
= 82 + 42
= 64 + 16
= 80
Square root both sides
C2 = 80
θ = tan-1 (2)
= 60
the angle between TC and tangent is 63º
TANGENT FROM A POINT
Suppose T is outside a circle there are two tangent from T to the circle and they are equal in length
Proof:
Consider the angle TCA and TCB
CA = CB (Both are radii)
TC = TC (Common)
TC = TC (Common)
CHORD AND TANGENT
Suppose the chord CD gets shorter and shorter is that C and D approach a common point E then the chord CD becomes the tangent at E, by interesting chord theorem.
XA x XB = AC x XD
XA x XB = XE x XE
XA x XB = (XE)2
Example
1.TX is a tangent to a circle. The line TAB cuts the circle at A and B with TA = 3cm and AB = 9cm. Find TX
Solution:
(TA) (TB) = TX x TX
(TA) (TB) = (TX)2
Since, TB = (TA) + (AB)
= 3cm + 9cm
= 12cm
From the theorem
(TA) (TB) = (TX)2
3cm x 12cm = (TX)2
Square root both sides
36cm2 = (TX)2
TX = ± 6cm
Since there is no -ve dimension therefore TX is 6cm
More Examples:
Find the length of unknown in the diagram.
Solution:
(CB) (CA) = (DC) (DC)
(CB) (CA) = (DC)2
Since CA = (CB) + (AB)
= 2m + a
From the theorem.
(CB) (CA) = (DC)2
(2m) (2m +a) = (4m)2
4m2 + 2ma = 16m2
2ma = 16m2 – 4m2
2ma = 12m2
a = 6m
a = 6m
Solution:
(CB) (CA) = (CD)2
Since CA = (CB) + (AB)
= 9m + 7m
= 16
From the theorem
(CB) (CA) = (CD)2
(9m) (16m) = (b)2
Square root both sides
144m2 = b2
b = ±12m
Since there is no -ve dimension b= 12m
2.TC is a tangent to a circle and Tab cuts at AB and B. if TA = 2cm and TB = 8cm, find TC
Soln:
From the theorem
(TA) (TB) = (TC)2
2cm x 8cm = (TC)2
16cm2 = (TC)2
Square root both sides
16cm2 = (TC)2
TC = 4cm
3. TX is a tangent to a circle and TYZ cuts the circle and Y and Z. if TX= 10m and TY = 4m. Find TZ
Solution:
(TY) (TZ) = (TX)2
Let ZY = y
Since TZ = TY + ZY
= 4m + y
From the theorem
(TY) (TZ) = (TX)2
(4m) (4m + y) = (10m)2
16m2 + 4my = 100m2
4my = 100m2 – 16m2
4my = 84m2
Y = 21m
Since ZY = y
TZ = TY + ZY
= 4m + 21m
= 25m
Class Activity
1.TA is a tangent to a circle and TBC meets the circle at B and C. TA = (9cm and BC = 24cm). Find TB
Solution:
(TB) (TC) = (TA)2
Let (TB) = y
Since TC = TB + CB
= y + 24cm
From the theorem
(TB) (TC) = (TA)2
(y) (y+ 24cm) = (9cm)2
Y2 + 24cmy = 81cm2
Y2 + 24cmy – 81cm2 = 0
By using General formula
Where a = 1, b = 24, c = 81
Since there is no negative dimension, the length at TB is 3cm
2.TX is a tangent to a circle and TPQ meets the circle at P and Q. TX = 12cm and PQ = 7cm, find TP
Solution:
(TP) (TQ) = (TX)2
Let (TP) = Z
Since TQ = TP + QP
= Z + 7cm
From the theorem
(TP) (TQ) = (TX)2
(z) (z +7cm) = (12cm)2
z2 + 7z = 144
z2 + 7z – 144 = 0
By using the general formula
Where a = 1, b = 7 and c = -144
Since there is no negative dimension, the length at TP is 9cm
3.AB is a chord at a circle at length 5cm. C is another point on the circle. AB extended on the circle meets the tangents at C and T. if the TC = 6cm, find the possible value of TB.
Solution:
(TB) (TA) = (TC)2
Let
(TB) = x
Since TA = TB + AB
= x + 5cm
From the theorem
(TB) (TA) = (TC)2
(x) (x + 5) = (6cm)2
X2 = 5x = 36
X2 + 5x – 36 = 0
By completing the square
X2 + 5x – 36 = 0
X2 + 5x – 36 = 0
X2 + 5x = 36
Add (½ b) 2 both sides
X2 + 5x + (½ x 5) = 36 + (½ x 5)2
X2 + (5/2)2 = 36 + 25/4
(x + 5/2)2 = 169/4
Square root both sides
(x + 5/2)2 = ±169/4
X + 5/2 = ± 13/2
X = -5/2 ± 13/2
= -5/2 + 13/2
= 4cm
Or
X = -5/2 – 13/2
= -9cm
Since there is no negative dimension, the length of TB is 4cm.
4. XY is a chord of a circle at length 2cm. z is another point on the circle. XY extended meets the length at z at T. if TX = 18cm, find the possible value of TZ
Solution:
(TY) (TX) = (TZ)2
Since TX = TY + XY
18cm + 2cm= 20cm
From the theorem
(TY) (TX) = (TZ)2
(TY) (TX) = (TZ)2
(18cm) (20cm) = (TZ)2
360cm2 = (TZ)2 square root both sides
(TZ) 2 = 360cm2
ALTERNATE SEGMENT THEOREM
AT is a tangent to the circle and AB is a chord. The alternate segment theorem state that:-
THEOREM: The angle between the chord and tangent is equal to the angle in the alternate (others)
Segment i.e. < TAB = 
Proof:
< TAB =
Aim: Is proving that
Let ACB = X and Centre of a Circle to be
‘O’ AOB = 2x (< at the centre is twice the angle at the circumference).
AC,BC, and AB or chords in the Circle and AO = OB
∴Δ AOB is isosceles triangle since AO and OB are equal
= 900 – x
=90° – (90° – x)
= 900-900 +x
=0+x
=0+x
∴
hence proved
∴
=x
hence proved
∴
=x
QUESTIONS:
1.ABCD is a cyclic quadrilateral TA is the tangent to the Circle at A. if 
0 = 1800 0 – 730
Solution:
TAC = 730
TAC = 730
Angle at the same segment equal
TAC = ADC
Take out 730 both sides
2. PQRS is a cyclic quadrilateral. ST is the tangent to the Circle at S. if 
0
=48º
Soln:
1320 +1320=180º
Take out 1320 both sides
=48º
3. C is the centre of the Circle. If 
Soln:
Angle at the Centre is twice the angle at circumstances
39º=< ABC
< ABC=39º x 2
=78º
=78º
4.X is the Centre of the Circle if
Soln:
Angle at the Centre is twice the angle at the circumstances.
∴
-Whereby O is the center of the earth.
-R is the radius of the earth
-The earth rotates once a day about a line called polar axis (earth axis)
-This axis basses through the center and joins the northern and southern poles
The distance of points A and B is 1223km
(b) 
The distance of point BC is 7891
The distance of point AC is 7,891km.
Since R is due to North of Q,the latitude of R is 20.33º
(a)Solution:
The distance between Tanga and Ruvuma is 
= 420.1Nm
The distance between point A and B is 370Km
The distance between two towns is 3929km.
664.3kmThe distance between Tabora and Tanga is 664km.
2.73 X 1000=2073Km The distance from town A to B is 2073km.
The speed of sheep is 691km/hour. 
510.7 km is the distance from point Q to R.
The journey took = 18 hours and 12 minutes
The distance from Zanzibar to Mtwara is 333.4km
=129.64km/hr
≈130km/hr
=60hours
∴
= 44º
THE EARTH AS A SPHERE
SPHERE
Is a set of a point which equidistance (equal distance) from the fixed point called the centre of the Sphere.
-The distance from the centre of the sphere to any point at the circumference of the sphere called Radius at the earth which is approximately as 6370km.
-The surface of the earth is not exactly spherical because it is flattened in its northern and southern pole or we say. The earth is not perfect sphere, as it is slightly flatter at the north and southern pole than at the equator. But for most purpose we assume that it is a sphere
THE EARTH AS PERFECT SPHERE
We consider the earth to be a perfect sphere of radius 6370km at approximately 6400km.
-Whereby O is the center of the earth.
-R is the radius of the earth
-The earth rotates once a day about a line called polar axis (earth axis)
-This axis basses through the center and joins the northern and southern poles
The equator has 00
DEFINITIONS OF TERMS
I. GREAT CIRCLE (EQUATOR)
Is the line which drawn from west to east with 00. Or
Is an imaginary line which divides the earth surface into two equal parts: Southern part and Northern part through the Centre of the earth called hemisphere.
The equator is the only Great Circle perpendicular to the earth axis
-Earth point on the earth’s surface is said to be either in northern hemisphere or southern hemisphere.
II. SMALL CIRCLE (LATITUDE)
Is the line drawn from West to east and measure in degree from the centre at the Earth (Equator) Northward or Southward.
– Latitude range from 00N or 900S
– The radius of parallel at latitudes becomes smaller as one moves towards the southern or northern pole
Example of the line of latitude that should be drawn from west to east .
– The equator is the standard zero latitude from which other latitude are measured
– Any other line of latitude is named by the longitude basses through when rotating from the Equator to the line of latitude this angle is either north or south of the equator.
When naming a latitude it is essential to say whether it is north or south of the Equator.
III. MERIDIAN (LONGTUDE)
– These are lines drawn from north to south measured degree from the prime meridian westward or eastward.
– These circles are not parallel as they meet at the poles. These circles have radius equal to that of the earth and they are called great circle.
– Longitudes are also called meridian.
Diagram:
– In order to name lines at longitudes it has been necessary to choose a standard zero called the prime meridian.
This is a line at longitude which passes through Greenwich, London.
IV.THE GREENWICH MERIDIAN
-Greenwich meridian: Is a longitude whose degree measure is zero (00)
-Greenwich meridian is also considered as prime meridian
-Greenwich meridian is standard longitude in which other meridian are measured in degree from East for West.
– The Greenwich divides the earth’s into two parts eastern part and western part
– Each point on the earth surface is said to be either on the eastern part or western part.
Diagram:
– The longitude of a place varies from 00 along the Greenwich meridian to 1800E or 1800W. The Radius of all longitude are equal to that of the earth.
LOCATION OF POINTS ON THE EARTH SURFACE
1.O is the centre of the earth. The latitude of P is 500N and of Q is 400S
Solution:
Since the given point has been allocated into different hemisphere therefore the angle subtended by an arc PQ = 500 + 400= 900
2. Two towns are on the same circle of longitude. One town is 200N and other is 300S. What is the angle subtended by an arc of these two angles.
Solution:
Since the two towns has been found into different hemisphere the angle subtended by an arc of the two angle are
200 + 300 = 500
3. Two towns C and D line on the equator. The longitude of C is 700E and for D is 30º
E.
E.
What is the angle subtended by an arc.
Solution:
Since C and D are in the same hemisphere the angle subtended by an arc CD = 700 – 300
CD= 400
CD= 400
4. Two towns A and B are on the equator.
The longitude of A is 350E and the B = 720W. Find the angle subtended by an arc AB
Since the point given has been found in the different hemisphere the angle subtended by an arc AB = 720 + 350
= 1070
HOME WORK
1.Given that Morogoro is (70S, 380E) and Moscow is (560N, 380E). Find the angle subtended by the area which connect the two places at the centre of earth.
Solution:
Since the given places have been alocated in the different hemisphere therefore subtended by an arc = 560 + 70
= 630
2. What is the difference in longitude between Brazivile Congo (40S, 150E) and Mombasa Kenya( 40S, 400E)
Solution:
Since the given places have been allocated in the same hemisphere therefore the different between the two places = 400 – 150
= 250
DISTANCE BETWEEN TWO PLACES MEASURED ALONG GREAT CIRCLE
Note: Great circles means either the equator or lines of longtudes.
Consider two points P and Q both found on the equator.
Diagram:
O is the centre of the earth OP = OQ = Radius of the earth angle POQ is the central angle line PO = ‘l” is the length at arc PQ on the equator. θ is the difference in longitudes between points P and point Q.
Remember, If P and Q are on the same hemisphere. θ will be found by subtracting their respective longitudes and if P and Q are in different hemisphere, the value of θ will be obtained by taking their sum of the respective longtude.
θ = l
3600 = 2πR
Whereby, π= 3.14
R = 6370km
Example:
Three points A(00,140W), B(00,250W) and C(00, 460E) are on the Earth’s surface.
Calculate the length of the equator
(a)AB (b) AC (c) BC
The distance of points A and B is 1223km(b)
(c)
The distance of point BC is 7891HOME WORK
1.Three points are such that A(430N, 100E), B(160N, 100E) and C(280S, 100E). calculates the lengths at the following arcs measured along the longitudes.
(a) AB (b) BC (c) AC
Solutions:
The distance at point AB is 3000km.
(b)
The length at point BC is 6156km.
(c)
The distance of point AC is 7,891km.Home Work
1.Two towns R and Q are 2813km apart R being direction of North of Q. If the latitude Q is 50S, find the Latitude of R.
Solution:-
Data
Distance RQ = 2813km
R = Required
Q = 50S
Since R is due to North of Q,the latitude of R is 20.33º
Class Work
1.Calculate the distance between Tanga (50, 390E) and Ruvuma (12ºS, 39°E) in;
(a) Nautical mile
(b) Kilometre
(a)Solution:
(α ± β) 600 = Nm
(120 – 50) 600 = Nm
(120 – 50) 600 = Nm
70 x 600 = Nm
(b) 1Nm = 1.852km
420.1Nm =?
The distance between Tanga and Ruvuma is(a) 420. 1Nm
(b) 778km
1.Find the distance between point x and y given that X(340N, 1240E) and Y(410N,1240E)
(a)In nearest Nautical mile
(b)In nearest Kilometers
Solutions:
(
β) 600 = Nm
β) 600 = Nm(410 – 340) 600 = Nm
70 x 600 = Nm
= 420.1Nm
The distance between point X and Y is
(a)420 Nm
(b)793Km
Home Work
1.An air Craft took a height from town A(40N, 120E) moving southwards along a great circle for a distance of 2437Km. write the position of the town it landed.
Solution:
Distance = 2437Km
A = 40N, 120E
B = Required (5º 120E)
4° + β = 21.93°
β
= 21.93° – 4°
=17.930
= 21.93° – 4°
=17.930
The position of town of Landed is 17.930
DISTANCE BETWEEN TWO POINTS ALONG THE PARALLEL OF LATITUDE.
Except the equator, other parallels of latitudes are small circles. Simply because their radii are less than the radius of the earth
Consider the point P and Q both found on the same parallel of latitude, let say α0N and Q is the different in longitudes between P and Q.
R is the radius parallel latitude of α0N
Diagram
Length at arc PQ = θ
Circumference 0f small circle 3600
But; r is the radius of small circle.
But PQ is the distance between two points but θ = α ± β
Example
1.Calculate the distance between P(500N, 120W) and Q(500N, 260E)
Solution:
Home Work
1. Find the distance between point A(580N, 230E) and (580N, 400W)
Solution
The distance between point A and B is 370KmCLASS ACTIVITY
1. Calculate the distance from town P and Q along the parallel of latitude.
If P (230N,100E) and Q (230N, 540E)
Solution:
4.501 x 1000 =4501 x 1
= 4501km
The distance from P to Q is 4501 Km
2. Two Towns both on latitude 450S differ in longitude by 500. Calculate the distance between two towns measured along the parallel of latitude.
Solution:
3.929 x 1000 = 3929 x 1
=3929Km
The distance between two towns is 3929km.Example;
A plane flying at 595km/hour leaves Dar es Salaam (70S, 390E) at 8:00 am. When will it arrive at Addis Ababa at (90N, 390E)
Solution:
1.778 x 1000
1778km
The distance from Dar to Addis Ababa (Ethiopia) is 1778km
= 2:54
Since it spent 2:54 and left at Dar around 8:00 am now will reach Addis Ababa at 10:54am
CLASS ACTIVITY
1.An aero plane flies from Tabora (50S, 330E) to Tanga (5ºS, 39ºE) at 332 km/hour along parallel of latitude. If it leaves at Tabora at 3:00 pm. Find the arrival time at Tanga airport.
1.An aero plane flies from Tabora (50S, 330E) to Tanga (5ºS, 39ºE) at 332 km/hour along parallel of latitude. If it leaves at Tabora at 3:00 pm. Find the arrival time at Tanga airport.
Solution.
664.3km
The distance between Tabora and Tanga is 664km.
= 2 hours
Time arrival time at Tanga air port is 3:00pm + 2:00pm = 5:00pm
Home Work
1.A ship is Steaming in a eastern direction from town A to town B. if the position A is (320N, 1360W) and B is (320N, 1380W). What is the speed of sheep
if it takes 3hours from town A to town B.
if it takes 3hours from town A to town B.
Solution:
2.73 X 1000=2073Km
The distance from town A to B is 2073km.
The speed of sheep is 691km/hour.In solving problems involving speed at ships, a term known as knot is usually used. By definition a speed of one nautical mile per hour is called knot
Therefore
1knote = 1Nm/hour = 1.852km/hour
Therefore
1knote = 1Nm/hour = 1.852km/hour
Example: 1
When a ship is given 20knots is actually sailing at 20 nautical miles per hour or approximately 37 kilometer per hour.
When a ship is given 20knots is actually sailing at 20 nautical miles per hour or approximately 37 kilometer per hour.
Example: 2
A ship sails northwards to Tanga (50S,390E) at an average speed of 12 knots. If the ship starting points is Dar es Salaam (70S, 390E) at 12:00noon, when will it reach Tanga.
A ship sails northwards to Tanga (50S,390E) at an average speed of 12 knots. If the ship starting points is Dar es Salaam (70S, 390E) at 12:00noon, when will it reach Tanga.
Solution: 1
How did 37km obtained.
1knote = 1.852km/hour
= 2 x 18.52km/hr
= 37.04km
= Approximately 37km
Solution: 2
1knot = 1.852km/hr
12knot= x
1knot x = 12knots x 1.852km/hr
X = 10.09
= 10:00
∴From 12:00 noon adding 10hours will sail at Tanga at 10:00pm.
12knot= x
1knot x = 12knots x 1.852km/hr
X = 10.09
= 10:00
∴From 12:00 noon adding 10hours will sail at Tanga at 10:00pm.
Home Work
1. A ship is teaming at 15knots in western direction from Q to R. if the position of P is 40ºS, 178ºe and that of Q is 40ºS, 172ºE, how long will the journey take?
1. A ship is teaming at 15knots in western direction from Q to R. if the position of P is 40ºS, 178ºe and that of Q is 40ºS, 172ºE, how long will the journey take?
Solution:
510.7 km is the distance from point Q to R.
= 18:12
The journey took = 18 hours and 12 minutesClass Activity
1. A speed boat traveling from Zanzibar (60S, 450E) to Mwanza (90S, 450E) using 30knots left Zanzibar at 11:30am at what time did it reach at Mwanza?
Solution:
Solution:
The distance from Zanzibar to Mtwara is 333.4km= 5:54pm
Since it spent 5:54pm and left at Zanzibar at 11:30am. Now it will reach at Mtwara 5:24pm.
2. Find the time taken for a ship to sail from town P(800N, 600W) to town Q(600S, 600W) in 70knots
Solution:
=129.64km/hr
≈130km/hr
= 119:36
The ship will take 119hours and 36 minutes from town P to town Q
Class Activity.
1. A ship sails from A (00,200W) to B (100N, 200W) at 16 knots. If it leaves A at 8:00am on Tuesday when will it reach B?
Solution:
1knot = 1.852km/hour
16knots =?
= 16 x 1.852km/hour
= 29.632km/hour
30km = 1hour
= 37hours
37 – 24hour = 13hours
8:00 + 24hr = 8:00am
8:00am + 13hours = 9:00am
The ship will reach town B at 9:00 am on Wednesday
Home Work
A ship sails from point A (100S, 300W) to B(100N, 300W) at 20 knots. If it leaves point A at 12:00 midnight on Monday when will arrive at B?
A ship sails from point A (100S, 300W) to B(100N, 300W) at 20 knots. If it leaves point A at 12:00 midnight on Monday when will arrive at B?
Solution:
1kont = 1.852km/hour
20knot = x?
knot x = 20knot x 1.852km/hour
≈37km/1hour
2222km = x?
37km x = 2222km x 1hour
=60hours
Since 12:00pm on Monday the ship has spend 2 and ½ days where by it will arrive at B at 12:00 afternoon on Thursday.
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