Circle theorems: photographing the park gate
Walk a photographer round the edge of a circular park and compare the angle at which the gate is seen with the angle at the centre.
Core concept
The angle at the centre is twice the angle at the circumference standing on the same arc
Topic: Circle theorems
A circular park has a gate AB on its edge, and a statue at the centre O. A photographer standing at a point P on the edge wants the whole gate in the picture, so the camera must cover the angle: The amount of turn between two lines that meet, measured in degrees (°). All key words APB. From the centre the gate fills the angle AOB. Circle theorems link these angles: the angle at the centre is twice the angle at the edge, so every place on the big arc sees the gate at the same angle. If the gate is a diameter, the angle at the edge is always 90°. From the small arc the angle is the supplement of the big-arc angle, because opposite angles of a cyclic quadrilateral add up to 180°.
What you will learn
- Use: the angle at the centre is twice the angle at the circumference.
- Use: angles in the same segment are equal, and the angle in a semicircle is 90°.
- Use: opposite angles of a cyclic quadrilateral add up to 180°.
Before you start
- Angles in triangles; isosceles triangles.
Materials
- A round plate, a protractor and string
Equipment
- A compass for drawing circles
Procedure
- Drag the photographer P round the edge of the park.
- Tap − or + to make the gate narrower or wider (its angle at O).
- Record P at three places on the big arc and compare ∠APB with ∠AOB.
- Make the gate a diameter, and move P onto the small arc; record each.
How this simulation is simplified
- The park's edge is a perfect circle and P stays on it.