Angle subtended at the centre

OBJECTIVE

To verify that the angle Subtended by an Arc at the Centre of a Circle is Double the Angle Subtended by the same Arc at Any Point on the Remaining Part of the Circle.

PREREQUISITE KNOWLEDGE

Meaning of angle subtended by an arc

 In the following image  

                                                                              Image 1

O is the centre

AB is an arc

Arc AB subtends ∠AOB at the centre

and  ∠APB on the remaining part of the circle.

MATERIALS REQUIRED 

Tracing papers

Coloured glazed paper

Geometry Box

Scissors

Fevicol/gum

PROCEDURE

Take a rectangular cardboard of a suitable size and paste a white paper on it

Cut a circle of suitable radius from a coloured glazed paper and paste on the cardboard.Take two point 

A & B ,to get an  Arc  (Image 2)

                                                                          Image 2

Join the points A & B to the centre to get ∠AOB.

Take any point P on the remaining part of the circle .Join P to A & B to get ∠APB.

Make a cut-out of  ∠AOB 

                                                                              Image 3

Make two cut-outs of ∠APB by using tracing papers.

                                                                                Image 4

Place the two cut-outs of ∠APB on the cut-outs of ∠AOB, as shown in the  Image5

                                                                                Image 5

OBSERVATION

We observe that two cut-outs of ∠APB completely cover the cut-outs of  ∠AOB.

Therefore

                                            2∠APB=∠AOB