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

Central Angle

The Formula for the Length of an Arc

Derive the formula for the length of an arc

Circumference of Circle = PI x diameter = 2 PI x radius where PI =𝝅= 3.141592...

The Central Angle

Calculate the central angle

A central angle is an angle formed by two intersecting radii such that its vertex is at thecenter of the circle.

<AOB is a central angle. Its intercepted arc is the minor arc from A to B.

The Concept of Radian Measure

Explain the concept of radian measure

Radians are the standard mathematical way to measure angles. One radian is equal to the angle created by taking the radius of a circle and stretching it along the edge of the circle.

The radian is a pure mathematical measurement and therefore is preferred by mathematicians over degree measures. For use in everyday work, the degree is easier to work with, but for purely mathematical pursuits, the radian gives better results. You probably will never see radian measures used in construction or surveying, but it is a common unit in mathematics and physics.

Radians to Degree and Vice Versa

Convert radians to degree and vice versa

The unit used to describe the measurement of an angle that is most familiar is thedegree. To convert radians to degrees or degrees to radians, the following relationship can be used.

angle in degrees = angle in radians * (180/pi)

So, 180 degrees = pi radians

Example 1

Convert 45 degrees to radians

Solution

45 = 57.32*radians

radians = 45/57.32

radians = 0.785

Most often when writing degree measure in radians, pi is not calculated in, so for this problem, the more accurate answer would beradians = 45 pi/180 = pi/4

Example 2

Convert pi/3 radians to degree

degrees = (pi/3) * (180/pi)

degrees = 180/3 = 60°

Angles Properties

Circle Theorems of Inscribed Angles

Prove circle theorems of inscribed angles

Aninscribedangle is formed when two secant lines intersect on a circle. It can also be formed using a secant line and a tangent line intersecting on a circle. Acentral angle, on the other hand, is an angle whose vertex is the center of the circle and whose sides pass through a pair of points on the circle, therefore subtending an arc.In this post, we explore the relationship between inscribed angles and central angles having the same subtended arc. The angle of the subtended arc is the same as the measure of the central angle (by definition).

In the first circle,is a central angle subtended by arc. Angleis an inscribed angle subtended by arc. In the second circle,is an inscribed angle andis a central angle. Both angles are subtending arc.

What can you say about the two angles subtending the same arc? Draw several cases of central angles and inscribed angles subtending the same arc and measure them. Use a dynamic geometry software if necessary. Are your observations the same?

In the discussion below, we prove one of the three cases of the relationship between a central angle and an inscribed angle subtending the same arc.

Theorem

The measure of an angle inscribed in a circle is half the measure of the arc it intercepts. Note that this is equivalent to the measure of the inscribed angle is half the measure of the central angle if they intercept the same arc.

Proof

Letbe an inscribed angle andbe a central angle both subtending arcas shown in the figure. Draw line. This forms two isosceles trianglesandsince two of their sides are radii of the circle.

In triangle, if we let the measure ofbe, then angleis also. By theexterior angle theorem, the measure of angle. This is also similar to triangle. If we let angle, it follows thatis equal to 2y. In effect, the measure of the inscribed angleand the measure of central anglewhich is what we want to prove.

The Circle Theorems in Solving Related Problems

Apply the circle theorems in solving related problems

Example 3

An arc subtends an angle of 200 at the center of the circle of radius25cm.Find the length of this arc.

Solution

r =25cm, 𝜽=20°

The length of the arc is 8.73cm.

Example 4

An arc of length 5cm subtends 50° at the center of the circle, what is theradius of the circle?

l=5cm, 𝜽=50°, r=?

Chord Properties of a Circle

Chord Properties of a Circle

Identify chord properties of a circle

Imagine that you are on one side of a perfectly circular lake and looking across to a fishing pier on the other side. The chord is the line going across the circle from point A (you) to point B (the fishing pier). The circle outlining the lake's perimeter is called thecircumference. Achord of a circleis a line that connects two points on a circle's circumference.

To illustrate further, let's look at several points of reference on the same circular lake from before. If each point of reference (i.e. duck feeding area, picnic tables, you, water fountain, and fishing pier) were directly on this lake's circumference, then each line connecting a point to another point on the circle would be chords.

• The line between the fishing pier and you is now chord AC
• The line between the water fountain and duck feeding area is now chord BE
• The line between you and the picnic tables is chord CD

If we had a chord that went directly through the center of a circle, it would be called adiameter. If we had a line that did not stop at the circle's circumference and instead extended into infinity, it would no longer be a chord; it would be called asecant.

The Theorem on the Perpendicular Bisector to a Chord

Prove the theorem on the perpendicular bisector to a chord.

Proof of Theorem

The Theorem on Parallel Chords

Prove the theorem on parallel chords

Parallel chords in the same circle always cut congruent arcs. Parallel chords intercept congruent arcs.

• Construct a diameter perpendicular to the parallel chords.
• What does this diameter do to each chord? The diameter bisects each chord.
• Reflect across the diameter (or fold on the diameter). What happens to the endpoints?The reflection takes the endpoints on one side to the endpoints on the other side. It, therefore, takes arc to arc. Distances from the center are preserved.
• What have we proven? Arcs between parallel chords are congruent.

The Theorems on Chords in Solving Related Problems

Apply the theorems on chords in solving related problems

Example 5

The figure is a circle with centreO. GivenPQ= 12 cm. Find the length ofPA.

Solution:

The radiusOBis perpendicular toPQ. So,OBis a perpendicular bisector ofPQ.

Example 6

The figure is a circle with centreOand diameter 10 cm.PQ= 1 cm. Find the length ofRS.

Solution:

Tangent Properties

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

Distances along Great Circles

Distances along Great Circles

Calculate distances along great circles

Take two places X and Y on the same line of longitude, i.e. one place is due north of the other. Suppose X is due north of Y. When travelling north from Y to X, you travel along part of a circle of longitude that is you travel along an arc of the circle.

The diagram below shows two points A and B on the same circle of longitude.

In the figure above, the sector containing the arc AB subtending∅, is shown.

Recall the formula for the length of arc.

NB: Remember, to find the difference in latitudes, take account of whether the places are north or south of the equator. If they are all found in south or north, then subtract the latitudes. If one is south and the other North then add the latitudes.

Nautical miles

Navigation Related Problems

Solve navigation related problems

Example 5

Find the distance between Alexandria (31°N, 30°E) and Kigali (2°S, 30°E)

Solution

Example 6

A plane starts at (20⁰S, 30⁰E), and flies north for 4000 km. Find its new latitude and longitude.

Example 7

A plane flies north from (10°S, 30°E) to ((27°N, 30°E) taking a time of 3 hours.

Find its speed, giving your answer I both knots and kilometers per hour.

Exercise 3

Consider the following Questions.

Distances along Small Circles

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.

Navigation

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

MATHEMATICS FORM THREE OTHER TOPICS
FORM THREE MATHEMATICS STUDY NOTES TOPIC 1-2.

FORM THREE MATHEMATICS STUDY NOTES TOPIC 3-4.

FORM THREE MATHEMATICS STUDY NOTES TOPIC 5.

FORM THREE MATHEMTICS STUDY NOTES TOPIC 6-7.

FORM THREE MATHEMTICS STUDY NOTES TOPIC 8.

O'LEVEL MATHEMATICS NOTES 
FORM ONE MATHEMATICS STUDY NOTES

FORM TWO MATHEMATICS STUDY NOTES

FORM THREE MATHEMATICS STUDY NOTES

FORM FOUR MATHEMATICS STUDY NOTES