Saturday, September 26, 2020
Home All Posts FORM THREE MATHEMTICS STUDY NOTES TOPIC 6-7.

# FORM THREE MATHEMTICS STUDY NOTES TOPIC 6-7. TOPIC 6: CIRCLES

Definition of Terms
Circle, Chord, Radius, Diameter, Circumference, Arc, Sector, Centre and Segment of a Circle
Define circle, chord, radius, diameter, circumference, arc, sector, centre and segment of a circle
A circle: is the locus or the set of all points equidistant from a fixed point called the center.
Arc: a curved line that is part of the circumference of a circle
Chord: a line segment within a circle that touches 2 points on the circle.
Circumference: The distance around the circle.
Diameter: The longest distance from one end of a circle to the other.
Origin: the center of the circle
Pi(π):A number, 3.141592…, equal to (the circumference) / (the diameter) of any circle.
Radius: distance from center of circle to any point on it.
Sector: is like a slice of pie (a circle wedge).
Tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.
NB: Diameter = 2 x radius of circle
Circumference of Circle = PI x diameter = 2 PI x radius
A Tangent to a Circle
Describe a tangent to a circle
Tangent
is a line which touches a circle. The point where the line touches the
circle is called the point of contact. A tangent is perpendicular to the
radius at the point of contact.
Tangent Properties of a Circle
Identify tangent properties of a circle
A
tangent to a circle is perpendicular to the radius at the point of
tangency. A common tangent is a line that is a tangent to each of two
circles. A common external tangent does not intersect the segment that
joins the centers of the circles. A common internal tangent intersects
the segment that joins the centers of the circles.
Tangent Theorems
Prove tangent theorems
Theorem 1
If
two chords intersect in a circle, the product of the lengths of the
segments of one chord equal the product of the segments of the other.  Intersecting Chords Rule: (segment piece)×(segment piece) =(segment piece)×(segment piece)
Theorem Proof:   Theorem 2:
If
two secant segments are drawn to a circle from the same external point,
the product of the length of one secant segment and its external part
is equal to the product of the length of the other secant segment and
its external part. Secant-Secant Rule: (whole secant)×(external part) =(whole secant)×(external part)
Theorem 3:
If
a secant segment and tangent segment are drawn to a circle from the
same external point, the product of the length of the secant segment and
its external part equals the square of the length of the tangent
segment. Secant-Tangent Rule:(whole secant)×(external part) =(tangent)2
Theorems Relating to Tangent to a Circle in Solving Problems
Apply theorems relating to tangent to a circle in solving problems
Example 7
Two
common tangents to a circle form a minor arc with a central angle of
140 degrees. Find the angle formed between the tangents.
Solution
Two tangents and two radii form a figure with 360°. If y is the angle formed between the tangents then y + 2(90) + 140° = 360°
y = 40°.
The angle formed between tangents is 40 degrees.

TOPIC 7: THE EARTH AS THE SPHERE

Features and Location of Places
The Equator, Great Circle, Small Circles, Meridian, Latitudes and Longitudes
Describe the equator, great circle, small circles, meridian, latitudes and longitudes
Definition of latitude and longitude
The
Earth is not a perfect sphere, as it is slightly flatter at the north
and south poles than at the equator. But for most purposes we assume
that it is a sphere.
The position of any point on earth is located by circles round the earth, as follows:
The earth rotates about its axis, which stretches from the north to the South Pole.
Circles round the Earth perpendicular to the axis are circles of Latitude and Circles round the Earth which go through the poles are circles of Longitude or meridians.
Consider the following diagram Normally Latitude is defined relative to the equator, which is the circle of latitude round the middle of the Earth while Longitude is defined relative to the circle of longitude which passes through Greenwich in London (Greenwich meridian).
The
latitude of a position tells us how far north or south of the equator
it is while the longitude of a position shows us how far east or west of
the Greenwich meridian it is.
Latitude; If we draw a line from the centre of the Earth to any position P , then the angle between this line and the plane of the equator is the latitude of P. Longitude: This is the angle between the plane through the circle of any Longitude P and the plane of the Greenwich meridian . Latitude can be either North or South of the equator while Longitude can be either East or West of Greenwich.
When locating the latitude and longitude of a place we write the latitude first then longitude.
Example 1
Dar
es Salaam has latitude 7°S (i.e. 7° south of the equator) and longitude
39°E (i.e. 39° east of the Greenwich meridian). So Dar es Salaam is at
(7°S, 39°E).
NB;
Greenwich itself has latitude 51°N ( i.e. 51° north of the equator)and
longitude 0° (by definition). Johannesburg has latitude 26°S (i.e. 26
south of the equator) and longitude 28°E(i.e. 28°east of the Greenwich
meridian), therefore Johannesburg is at (26°S, 28°E).The north pole has
latitude 90°S but its longitude is not defined. ( Every circle of
longitude goes through the north pole).The south pole has latitude 90°s.
Its longitude is not defined. So all points on the equator (such as Nanyuki in Kenya) have latitude 0°
Ranges; Latitude varies between 90°S (at the south pole) to 90°N (at the north pole).
Ranges; Latitude varies between 90°S (at the south pole) to 90°N (at the north pole). Longitude varies between 180°E and 180°W. These are the longitudes on the opposite side of the Earth from Greenwich.
GREAT AND SMALL CIRCLES:
There is an essential difference between latitude and longitude.
Circles of longitude all have equal circumference. Circles of latitude
get smaller as they approach the poles. The centre of a circle of
longitude is at the centre of the earth. They are called great circles. For circles of latitude, only the equator itself is a great circle. Circles of latitude are called small circles.
Example 2
Find the latitudes and longitudes of A and B on the diagram below; Solution;
The point A is 60° above equator, and 20° east of Greenwich.
So the point A is at (60°N, 20°E)
The point B is 10° below the equator, and on the Greenwich meridian.
So the point B is at (10°S, 0°).
Exercise 1
1. Write down the latitude and longitude of the places shown on figures below: 2. Copy the diagram show on the figure above and mark these points:
1. (10°N, 30°E)
2. (20°N, 20°W)
3. (0°, 20°W)
3. Obtain a globe, and on it identify the following places.
1. (40°S, 30°E)
2. (50°S,20°W)
3. (10°N,40°W)
4. (40°N, 30°E)
5. (80°N,10°E)
6. (0°,0°)
Difference between angles of latitude or longitude
Suppose two places have the same longitude but different latitudes. Then they are north and south of each other.
In finding the difference between the latitudes take account of whether they are on the same side of the equator or not.
• If both points are south of the equator subtract the latitudes
• If both points are the north of the equator subtract the latitudes
• If one point is south of the equator and the other north then add the latitudes
Similarly, suppose two places have the same latitudes but different longitudes:
• If both points are east of Greenwich subtract the Longitudes
• If both points are west of Greenwich subtract the Longitudes
• If one point is east of Greenwich and the other west then add the longitudes
Suppose places A and B are on the same longitude, then the difference in latitude is theangle subtended by AB at the centre of the earth. Suppose places A and B are on the same latitude.
Then the difference in longitude is the angle subtended by AB on the earth’s axis. Locating a Place on the Earth’s Surface
Locate a place on the Earth’s surface
Example 3
Three
places on longitude 30°E are Alexandria (in Egypt) at (31°N, 30°E),
Kigali (in Rwanda) at (2°S, 30°) and Pietermaritzburg (in South Africa)
at (30°S, 30°E).
Find the difference in latitude between
1. Kigali and Pietermaritzburg
2. Kigali and Alexandria
Solution
(a)Both Towns are south of the equator. So subtract the latitudes. 30 – 2 = 28
Therefore the difference is 280
(b) Kigali is south of the equator, and Alexandria is north, so add the latitudes
31 + 2 = 33
The difference is 330
Example 4
A
plane starts at Chileka airport (in Malawi) which is at (16°S, 35°E).
It flies west for 50°. What is its new latitude and longitude?
Solution
Since it flies west, then subtract 35° from 50°. This gives 15°
The new longitude is now west of Greenwich, hence the plane is at (16°S, 15°W).
Exercise 2
1. In the diagram shown in the following figure find,
1. The difference in longitude between A and B
2. The difference in longitude between D and E  Distance along Small Circles
Calculate distance along small circles
Suppose
P and Q are places west or east of each other, i.e they lie on the same
circle of latitude. Then when you travel due east or west from P to Q
you travel along an arc of the circle of latitude.
The
situation here is slightly different from that of the previous section.
While circles of longitude all have the same length, circles of
latitude get smaller as they get nearer the poles.
Consider the circle of latitude 50°S. Let its radius be r km.  Nautical miles Example 8
Find the distance in km and nm along a circle of latitude between (20°N, 30°E) and (20°N, 40°W).
Solution:
Both places are on latitude 20°N. The difference in longitude is 70°. Use the formula for distance.
Distance = 111.7 cos20° x 70°. Hence the distance in nautical miles is 60 x 70 x cos20°
The distance is 3,950 nm.
Example 9
A ship starts at (40°S, 30°W) and sails due west for 1,000 km. Find its new latitude and longitude. Example 10
A ship sails west from (20°S, 15°E) to (20°S, 23°E), taking 37 hours. Find speed, in knots and in kms per hr. Exercise 4
Consider the following Questions. Suppose
a ship is sailing in a sea current, or that a plane is flying in a
wind. Then the course set the ship or plane is not the direction that it
will move in. the actual direction and speed can be found either by
scale or by the use of Pythagoras’s theorem and trigonometry.
Draw
the line representing the motion of the ship relative to the water. At
the end of this line draw a line representing the current. Draw the
third side of the triangle. This side, shown with a double – headed
arrow, is the actual course of ship.
Example 11
A
ship sets course due east. In still water the ship can sail at 15km/hr.
There is a current following due south of 4kkm/hr. use a scale drawing
to find.
1. The speed of the ship
2. The bearing of the sip.
Solution:
In
one hour the ship sails 15km east relative to the water. Draw a
horizontal line of length 15cm. In one hour the current pulls the ship
4km south. At the end of the horizontal line, draw a vertical line of
length 4cm. Example 12
The ship of example 10 needs to travel due east. Calculate the following.
1. What course should be set?
2. How long will the ship take to cover 120km?
Solution
The ship needs to set a course slightly north of east, consider the following diagram. Note:
With no current, the journey would take 8hrs. The journey takes
slightly longer when there is a current.Suppose a ship or a plane does
not directly reach a position. We can still find how close the ship or
plane is to the position. Example 13
A
small island is 200km away on a bearing of 075°. A ship sails on a
bearing of 070°.Find the closest that the ship is to the island. Exercise 5
1. Find the difference in longitude between Cape Town (34°S, 18°E) and Buenos Aires (34°S, 58°W)
2. A ship startsat (15°N, 30°W) and sails south for 2,500km. Where does it end up?
3. Find thedistance in km along circle of latitude between cape Town and Buenos aires (see question 2)
4. A plane starts at (37°S, 23°W) and flies east for 1,500 km. where does it end up?

5. Find thedistance in nau