Construction of direct common tangent

The direct common tangents to two circles meet on the line of centres and divide it externally in the ratio of the radii. 

Objectives

To construct direct common tangent for the given circle.

Estimated Time

90 minutes

Prerequisites/Instructions, prior preparations, if any

1. The students should have prior knowledge of a circle , tangent and the limiting case of a secant as a tangent.
2. They should understand that a tangent is always perpendicular to the radius of the circle.
3. They should know construction of a tangent to a given point.
4. If the same straight line is a tangent to two or more circles, then it is called a common tangent.
5. If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent.

Materials/ Resources needed

• Digital: Laptop, geogebra file, projector and a pointer.
• Non digital: Students' individual construction materials.
• Geogebra file: This geogebra file was created by Mallikarjun sudi of Yadgir.

Download this geogebra file from this link.

Process (How to do the activity)

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'

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

• Developmental Questions (What discussion questions)

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

• Evaluation Questions

1. Is the student able to comprehend the sequence of steps in constructing the tangent.
2. Is the student able to identify error areas while constructing ?
3. Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
4. 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

1. What do you think are the applications of tangent constructions ?
2. What is the formula to find the length of direct common tangent ?
3. Can a direct common tangent be drawn to two circles one inside the other ?
4. Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
5. What are properties of direct common tangents ?
6. [Note for teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.] Examine with the help of following geogebra file made by Ranjani.