LSC S1:TANGENTS TO CIRCLES

Introduction
Have you ever pushed a wheelbarrow, ridden a bicycle or even seen someone push a patient or elderly person in a wheelchair uphill? As the wheel moves, the wheel and the ramp/road surface have one point in common (when they are in contact) along the plane. That point is called the point of tangency.

Therefore the line that touches a circle at one point is called a tangent to a circle. The same can be said of the movement/action of the chain as you pedal a bicycle. The point where the chain and free wheel touch each other is a point of tangency.

Tangents to circles have a number of applications in the design of different items such as pulleys, conveyor belts, vehicle mechanical systems and design of products that have curved forms.
Have you ever used or seen an escalator? Have you ever seen how bags on airport checkpoints are moved using cargo conveyors? Have you ever looked inside an open bonnet of a car and observed the belts? Have you
ever seen people at a construction site moving loads from the ground to upper floors using pulley systems? All these have elements of tangents to circles in their designs. In this chapter, you will identify, construct and apply tangents in design and drawing.

Tangents
Tangents have a number of applications in design as you have seen in the

What you need: 2H pencil, set, ruler, pair of compasses, dividers, eraser, T-square, board, clips, a computer or smart phone with an internet connection where possible and Geometrical textbook.
What to do: In pairs or groups, carry out this activity.

1. Describe each of the following tangents to circles. Use sketches and identify where they are used/applied in real life. It could be in simple machines or object designs.
   A tangent to a circle from a point outside
   A common external tangent to two equal circles
   A common internal tangent to two equal circles
   A common internal tangent to two unequal circles
   A common external tangent to two unequal circles
   Circles tangentially touching externally
2. Present your findings to your classmates in your respective groups for feedback.
3. Write down what you have learnt from other groups.

Construction of Tangents
It is important to be able to accurately construct different tangents to circles. They are very common in designs of products with curved shapes and parts of machines and systems.

Constructing a Tangent to a Circle

Constructing a Line Tangential to a Given Circle from a Point Outside the Circle

Constructing an External Tangent to Two Equal Circles

5.7 Constructing an Internal Tangent to Two Equal Circles
Activity 5.5
Constructing an internal tangent to two equal circles
What you need: 2H pencil, set, ruler, pair of compasses, dividers, eraser, T-square, board, clips and Geometrical textbook
What to do: Individually, carry out this activity.
Construct an internal tangent to two equal circles below.

Constructing a Common External Tangent to Two Un-Equal Circles
Which type of tangents to circles are used in the design of a bicycle chain and chain wheel?

Activity 5.6
circles
What you need: 2H pencil, set, ruler, pair of compasses, dividers, eraser T-square, board, clips and Geometrical textbook What to do: Individually, carry out this activity.
Draw two unequal circles and construct a common external tangent as in
Figure 5.8 below.

5.9 Constructing an Internal Tangent to Two Unequal Circles
O Activity 5.7
Constructing an internal tangent to two unequal circles What you need: 2H pencil, set, ruler, pair of compasses, dividers, eraser,
T-square, board, clips and Geometrical textbook What to do: Individually, carry out this activity.
Draw the two circles of radius 30 mm and 20 mm with centres A & B 80 mm apart and construct an internal tangent to them.

5.10 Constructing a Tangential Circle to an Angle
Activity 5.8
Constructing a tangential circle in an angle
What you need: 2H pencil, set, ruler, pair of compasses, dividers, eraser, T-square, board, clips and Geometrical textbook.
What to do: Individually, carry out this activity.
Draw the given angle PQR = 450. PQ = 76 mm and QR = 70 mm. Given that the circle to be drawn has a radius of 16 mm.
Draw a tangential circle of radius 16 mm to this angle.

Constructing Two Circles Tangentially Touching Externally Circles tangentially touching each other are common in engines.

5.12 Pulleys
Pulleys are good examples where tangents to circles are applied.
A pulley is a simple machine that uses grooved wheels and a rope or cable to raise or lower loads. There are three main types of pulleys: fixed, movable, and compound.
Pulleys have a number of applications in society such as running maize, rice and coffee mills and lifting materials.

I. Describe where a pulley is used in each item or machine in the following pictures
(Figure 5.13 and Figure 5.14).
2 List the names of items or machines used in the activities shown in the respective pictures.
3 State where else you have seen pulleys being used in your community.
4 Record your findings in a chart in your respective groups and present to the rest of the class for comparison and feedback.

Making a Mock-up of a Fixed Pulley
Making mock- ups of pulley systems will help you understand the application of tangents in designs of pulleys.

Activity 5.11
Making a mock-up of a fixed pulley What you need: 2H pencil, set, ruler, compass, eraser, board, a computer with an internet connection where possible, modelling material such
as clay, nylon string or thread and a round stick, tools such as a cutting
blade, measuring tools such as steel rule or tape measure.

What to do: In groups, carry out this activity.

1. Make a mock-up ofa fixed pulley you would use for raising heavy loads.
2. Sketch your designs for the pulley system.
3. Construct the final design of the pulley system with dimensions.
4. Using your final drawings, make mock-ups of the pulley system.
5. Present your mock-ups to the class and explain how they work.
6. Improve your mock-ups according to the feedback you get.
7. Apply safety precautions.
   Different pulley designs have different tangents as you will discover in the activities
   Conveyor Systems
   Conveyor systems are the other examples where tangents to circles are used in design. A conveyor system is a fast and efficient mechanical handling method for automatically transporting loads and materials within a given area. These systems use belts, wheels, rollers, or chains to transport objects.
   Conveyor systems are used in warehouses and factories, lifts, escalators, machines etc.

What you need: 2H pencil, set, ruler, compass, eraser, board, clips, a computer with an internet connection where possible. What to do: In groups, carry out this activity.

1. Where are the respective conveyor systems used in Figure 5.16 above?
2. What are the functions of the respective conveyors?
3. In the absence of conveyors, what other means are used in each respective case to serve the same function?