### Center Path

Four rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of the point X and prove your assertion.

### Tied Up

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal can reach all points in the field. Which one is it and why?

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

# Right Angles

### Why do this problem?

This problem builds on Triangles in Circles and Subtended Angles . It leads to the theorem about right-angles in circles.

Teachers may find the article Angle Measurement: An Opportunity For Equity to be of interest.

### Possible approach

If students are going to be working at individual computers, demonstrate how the geoboard works - clicking on a coloured rubber band, dragging it onto a peg and then "stretching" it out onto two more pegs to make a triangle.

If students will be working on paper ask them to draw right-angled triangles on their 9-peg and 12-peg circles.

How many different triangles can they find?
How do they know they are right-angled?
What is special about the right-angled triangles?

Draw together conjectures which might mention the number of dots on the circle and the need to be able to join two points to form a diameter. Challenge students to justify these conjectures with convincing arguments. Eventually link this to the work on Subtended Angles .

### Key questions

What do we know already that might be useful here?
What is the relationship between the angle at the centre and the angle at the circumference?
What are the implications of our findings for circles in general (without dots)?

### Possible extension

Ask students to prove the general case.

### Possible support

Students may need to spend more time convincing themselves when it is possible to make right-angled triangles. Here is a sheet with 10-peg, 12-peg and 16-peg circles to support them in investigating other cases.