# FORM THREE MATHEMTICS STUDY NOTES TOPIC 6-7.

**TOPIC 6: CIRCLES**

**Definition of Terms**

A Tangent to a Circle

Describe a tangent to a circle

Tangent Properties of a Circle

Identify tangent properties of a circle

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**

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*
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.

- The speed of the ship
- 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.

- What course should be set?
- 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.

**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.**

*Note:*
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?

**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**
## No comments