# Circles Constructions

From Karnataka Open Educational Resources

Philosophy of Mathematics |

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## Contents

- 1 Concept Map
- 2 Textbook
- 3 Additional Information
- 4 Teaching Outlines
- 5 Hints for difficult problems
- 6 Project Ideas
- 7 Math Fun

# Concept Map

# Textbook

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# Additional Information

## Useful websites

## Reference Books

# Teaching Outlines

## Concept #1. Construction of chord.

### Learning objectives

### Notes for teachers

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

## Concept #2. Construction of Tangent.

### Learning objectives

### Notes for teachers

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

## Concept #3. Construction of direct common tangent.

### Learning objectives

### Notes for teachers

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No # Construct a direct common tangent to two circles with given radii and given distance between the centre of two circles.

- Estimated Time: 90 minutes
- Materials/ Resources needed: # Laptop, geogebra file, projector and a pointer.

- Students' individual construction materials.

- Prerequisites/Instructions, if any:

- The students should have prior knowledge of a circle , tangent and the limiting case of a

secant as a tangent.

- They should understand that a tangent is always perpendicular to the radius of the circle.
- They should know construction of a tangent to a given point.
- If the same straight line is a tangent to two or more circles, then it is called a common tangent.
- If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent.
- Note: In general,

- The two circles are named as C1 and C2
- The distance between the centre of two circles is 'd'
- Radius of one circle is taken as 'R' and other as 'r'
- The length of tangent is 't'
- Multimedia resources: Laptop
- Website interactives/ links/ / Geogebra Applets
- Process:

[Note for teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.]

The teacher can explain the step by step construction of Direct common tangent and with an example

- Developmental Questions:

- What is a tangent
- What is a common tangent ?
- What is a direct common tangent ?
- What is R and r ?
- What does the length OA represent here ?
- Why was a third circle constructed ?
- Let us try to construct direct common tangent without the third circle and see.
- What should be the radius of the third circle ?
- Why was OA bisected and semi circle constructed ?
- What were OB and OC extended ?
- What can you say about lines AB and AC ?
- Name the direct common tangents .
- At what points is the tangent touching the circles ?
- Identify the two right angled triangles formed from the figure ? What do you understand ?

- Evaluation:

- Is the student able to comprehend the sequence of steps in constructing the tangent.
- Is the student able to identify error areas while constructing ?
- Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
- Is the student able to appreciate that the direct common tangents from the same external point are equal and subtend equal angles at the center.

- Question Corner:

- What do you think are the applications of tangent constructions ?
- What is the formula to find the length of direct common tangent ?
- Can a direct common tangent be drawn to two circles one inside the other ?
- Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
- What are properties of direct common tangents ?

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

## Concept #4. Construction of transverse common tangent.

### Learning objectives

### Notes for teachers

### Activity No #

- Estimated Time
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- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

### Activity No #

- Estimated Time
- Materials/ Resources needed
- Prerequisites/Instructions, if any
- Multimedia resources
- Website interactives/ links/ / Geogebra Applets
- Process/ Developmental Questions
- Evaluation
- Question Corner

# Hints for difficult problems

# Project Ideas

# Math Fun

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