# Triangles in circles

Can you find triangles on a 9-point circle? Can you work out their angles?

## Problem

*Triangles in Circles printable sheet**9 dot circles printable sheet*

In the GeoGebra interactivity below there is **a circle with 9 equally spaced points on the edge**, and one in the centre.**Draw as many different triangles as you can, by joining the centre dot and any two of the dots on the edge.**

Can you work out the angles in your triangles?

You should have found four different triangles with angles of:

80, 50, 50

120, 30, 30,

160, 10, 10

**Here is a triangle formed by joining three dots on the edge of the nine-point circle:**

**Can you work out the angles of this triangle?***Click to reveal a diagram that might help you work out the angles.*

Create some more triangles by joining three dots on the edge of the nine point circle.**How many different triangles can you make?****Can you work out the angles each time?**

When the centre dot isn't inside your triangle, you might find it a little trickier to work out the angles. Click below for a diagram that might help:

*Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board.*

## Student Solutions

Arielle and Ethan from Emanuel School in Australia considered drawing all of the possible triangles in the circle. Arielle said:

To solve this problem, I went to each dot and joined it with all of the other remaining dots, starting with the middle dot and then going clockwise around the circle. This means that you have connected all of the dots and that it is impossible to create anymore triangles.

As all of the triangles are on the same diagram, it is difficult to tell which is which and to count how many there are. Lexie from Walton High School in England used a systematic method to count the triangles. However, Lexie counts some of the triangles more than once - can you work out which ones?

You can make 36 triangles in the circle. This is because there are 9 triangles with 0 dots in between (three examples are shown below)

and 9 triangles with 1 dot inbetween. (two examples are shown below)

There are also 9 triangles with 2 dots inbetween and 9 with 3 dots inbetween. However there is no point in doing triangles with 4 dots inbetween because on the other side of the circle there is 3 dots so you would be counting those triangles for the second time. The same goes for 5,6 and 7 dots inbetween. Therefore, if there are 4 sets of 9, you can make 36 triangles in the circle.

Thanima from Eastlea Community School in the UK looked at the angles at the centre of the circle, as in the diagrams shown below:

When the band is changed form one peg to the next the difference in the angle at the centre will be 40 degrees:

Click here to see Soham's work.

Lily from NLCS Jeju in South Korea noticed that she could do $180\div 9$ and then could find angles at the edge by doing $180\div9$ multiplied by how many 'segments' the opposite side used.

Click here to see Lily's work.

Why does Lily's method work?

This diagram shows an angle opposite a 1-segment side in a red triangle, which Lily says is 20 degrees, and an angle at the centre in a green triangle, which Thanima showed is 40 degrees.

These two triangles share a base, and one of them has the other angle at the centre while the other has its other angle on the circumference. There is a special relationship between these two angles!

Can you discover the relationship by finding these angles for other examples?

Can you discover, or explain, the relationship using a diagram like the one below?

And what about if the red triangle doesn't contain the centre?

## Teachers' Resources

### Why do this problem ?

This problem challenges students to apply what they know about angles in triangles. The nine-point circle allows students to concentrate on the geometrical structure while keeping the arithmetic reasonably straightforward.

It offers a good preparation for the problems Subtended angles, Right angles, and Cyclic Quadrilaterals, which lead to proofs of circle theorems.

Teachers may find the article Angle Measurement: an Opportunity for Equity of interest.

### Possible approach

*"How many different triangles can you make which consist of the centre point and two of the points on the edge?* "*"Can you find all the angles of the triangles?"*

At some stage students will want to know what makes triangles "different". This is a good opportunity to talk about congruence.

After a short time bring the class together to discuss how they calculated the angles.

Next, draw a triangle with three vertices on the edge of the circle which encloses the centre point, such as the one illustrated here.

*"Using what you already know, can you find all the angles of this triangle?"*

Once students have a strategy, invite them to find as many different triangles as they can, and work out the angles. At some stage, you may need to discuss how to cope with triangles that do not enclose the centre. There is a hidden diagram in the problem that may be helpful for this.

### Key questions

What do you know that might be useful?

Are there any extra lines you can draw, to help make the problem more like something you've seen before?

### Possible extension

Subtended angles, Right angles, and Cyclic Quadrilaterals are good follow-up problems that lead on to proofs of circle theorems.

### Possible support

Some students may benefit from working on some of the Angles, Polygons and Geometrical Proof Short Problems in preparation for this task.