Is there anything intuitive about tangents drawn to a circle from the same point outside of a circle?

I sent this Quick Poll to my students without telling them anything.

I watched and listened as students talked with each other.

They drew diagrams, some to scale, and some not, and decided that the tangent segments drawn to the circle from the same point outside the circle were congruent.

How do you know?

BK and his team knew for sure because they constructed the diagram using dynamic geometry software.

Why do the tangent segments have to be congruent?

Students practiced **look for and make use of structure**. What do you see that isn’t pictured?

Many students drew in some diameters.

What do you see that isn’t pictured?

Some students recognized that a radius drawn to a point of tangency will be perpendicular to the tangent.

What do you see that isn’t pictured?

A kite!

Why is it a kite?

The radii are congruent.

Segment AC is congruent to itself.

The triangles are right, so angles B and D are congruent.

We got ∆ABC congruent to ∆ADC by HL.

Then the tangent segments are congruent because the triangles are congruent.

And then back to the dynamic geometry software to make more sense of the diagram we had been given.

What kind of #AskDontTell opportunities are you providing the learners in your care this week?

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howardat58

March 15, 2015 at 8:09 pm

It is interesting that none of them thought about the centre of the circle and the lin from the centre to the point, followed by an argument from Symmetry.

jwilson828

March 16, 2015 at 5:57 am

That is interesting. We talked a lot about symmetry of the kite in our unit on Polygons, but that’s just not what this year’s group thought of during this lesson.

mrdardy

March 16, 2015 at 4:30 am

Jennifer – you mention that some students recognized that a radius drawn to a point of tangency forms a right angle. Can you say more about this? I feel that this is the key idea but I also feel a bit like I am applying, um, circular logic (pardon the pun) when I try to explain why this perpendicular relationship holds true. I’d love to hear how they discovered this or how you explain it.

Thanks for another fun post!

howardat58

March 16, 2015 at 6:55 am

Symmetry can be appealed to on this one as well. If you reflect the picture in the radius line and the tangent line did not reflect to itself, but its reflection is still a tangent, then you have two distinct tangents at a point of the circle – IMPOSSIBLE !

jwilson828

March 16, 2015 at 7:47 am

Thank you for always pushing us to think about geometry from a transformational point of view. My students help me do that more than I do naturally, but you always remind us when that’s not how we see the geometry unfold.

jwilson828

March 16, 2015 at 7:45 am

I know exactly what you mean. Every year before the radius drawn to a tangent is perpendicular, I think back through how to prove it myself. We don’t spend a lot of time on this proof – we just don’t have time to prove every result in detail. We use the technology to recognize that it happens (starting with a radius drawn to a secant line & then moving the secant line into a tangent line, observing what happens), and then we talk through a proof of why it has to be true. We use a proof by contradiction to show it must be true.

If you haven’t already, you might want to read through this IM task, which has several good proofs as to why it must be true.

https://www.illustrativemathematics.org/content-standards/tasks/963

howardat58

March 17, 2015 at 4:46 am

I had a look at that. They make a real meal of the transformation proof. In fact one can argue that the “tangent is perpendicular to the radius” and “there is only one tangent at a point” are equivalent statements about a circle, and so it is pushing it to use one to “prove” the other.

I am unsure as to the need for formality in proofs at this level in math, the CCSS does not make this clear at all. How about this: “If there were two tangents to the circle at a point then the circle would have a vertex, or a cusp, at that point, but it doesn’t!”. Is this sort of argument too informal?