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.

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.

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?

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

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

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

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.

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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Circle Theorems with Geogebra – Part 1

There are various applets available online and indeed good Geogebra constructions that seek to demonstrate the various Circle Theorems.

However, I think it is more powerful to construct them from scratch in front of the class using Geogebra, starting from a blank sheet. It requires a degree of familiarity with Geogebra and reliable IT infrastructure (something which my school sadly lacks) but I think it is a more instructive way of walking through them than just moving some points on an existing animation.

So, this blog post, is an attempt to show my progression through the Circle theorems.

1, Set up.

First, get the Geogebra page set up, remove the grid and the axes as they are distracting. Also change the menu options so your screen isn’t littered with labels.

2, Draw a circle!

Point a point for the centre and then draw a nice big circle and point a couple of points on the circumference, talking through this as you are doing it using the terms in bold.

3, Angles subtended by the same arc are equal

You need three points, but best not to use the one which was used to define the circle as this will change the size of the circle. Two of the points define an arc.  Then draw two line segments to subtend an angle from that arc.

Then measure the angle between the lines and show how it changes:

Maybe add another point to create a second angle subtended by the same arc. Ask students to predict what the angle will be before measuring it.

You might want to highlight the arc itself using the arc tool.

4, The angle subtended by a semi-circle is 90° OR The triangle in a semi-circle is right-angled

Next investigate what happens as you adjust the arc and look at the special case that occurs when the angle is 90°.

5, The Angle subtended at the centre is double the angle subtended at the arc

Now add two more line segments to create an angle at the centre (deleting the line you drew previously).  Measure that angle and ask students what they notice as you change the arc.

You might want to have another look at the special case of when the arc is a semi-circle and reason why the angle at the circumference is 90°. And indeed show that this works for reflex angles too.

An alternative way of discovering this theorem is to construct this special case and then reason that because you have an isosceles triangle formed by two radii, you can show the the other angle in the isosceles triangle will be 180 – 2x the base angle. Therefore the other angle at the centre will be 2x the base angle.

We’re not there yet, check back for Part 2 where we will go outside the circle.  Oooohh….