Construction of direct common tangent

=Activity - Construction of direct common tangent=

Estimated Time
90 minutes

Materials/ Resources needed

 * 1) Laptop, geogebra file, projector and a pointer.
 * 2) Students' individual construction materials.

Prerequisites/Instructions, 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'

Multimedia resources
Laptop

Website interactives/ links/ simulations/Geogebra Applets
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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 for assessment of the child)

 * 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.

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Activity Keywords

 * 1) Geogebra
 * 2) Direct Common Tangent

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