Construction of direct common tangent

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.
 * 6) 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'

Materials/ Resources needed
This geogebra file was created by Mallikarjun sudi of Yadgir.
 * Laptop, geogebra file, projector and a pointer.
 * Students' individual construction materials.

Process (How to do the activity)
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.