Circle Theorems with Geogebra – Part 2

This is the second post in this series.  Part 1 is here.

 

1, Tangents

What is a tangent?  If you look at a line and a circle, it can either not touch the circle at all, cut it in two places, or if we are really careful, just touch it in exactly one place.

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The special thing about a tangent is that there is only one line that can form a tangent at any point on the circle. If you move the line then the point of contact also moves.  Geogebra works by drawing both tangents from a point to the circle.  By putting points at the intersect you can see how they move.

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As you connect the points on the circumference to the centre, stop to ask what we know about the lines that have been drawn here.  Then what do we notice about the two tangent lines and radius as we move them?

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To prove the point we can measure the angle between the radius and the tangent:

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And measure some lines too, whilst moving things around a bit:

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We can deduce that DE = FD, by drawing in one more line and proving that we have congruent triangles AED and AFD.

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2, Cyclic Quadrilaterals

What is a quadrilateral? How many lines, how many vertices? So what is a cyclic quadrilateral? Once you’ve established this, draw one.

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Then go round measuring the angles. Before you move one of the points, ask students to predict what will change and what will stay the same.

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