Cambridge International Mathematics

7 In this question we prove thetangents from an external point

theorem.

a Join OP, OA and OB.

b Assuming thetangent-radiustheorem, prove that¢s POA

and POB are congruent.

c What are the consequences of the congruence inb?

THEOREMS INVOLVING ARCS

Any continuous part of a circle is called anarc.

If the arc is less than half the circle it is called aminor arc.

If it is greater than half the circle it is called amajor arc.

A chord divides the interior of a circle into two regions called

segments. The larger region is called amajor segmentand the

smaller region is called aminor segment.

In the diagram opposite:

² the minor arc BCsubtendsthe angle BAC, where A is on

the circle

² the minor arc BC also subtends angle BOC at the centre of

the circle.

Discovery 2 Theorems involving arcs

#endboxedheading

The use of thegeometry packageon the CD is recommended, but the discovery can also be done using

a ruler, compass and protractor.

Part 1: Angle at the centre theorem

What to do:

1 Use a compass to draw a large circle with centre O. Mark

on it points A, B and P.

2 Join AO, BO, AP and BP with a ruler.

Measure angles AOB and APB.

3 What do you notice about the measured angles?

4 Repeat the above steps with another circle.

5 Copy and complete:

“The angle at the centre of a circle is ...... the angle on

the circle subtended by the same arc.”

a minor arc BC

B C

a major arc BC

B C

A

B C

O

major segment

minor segment

P

A

B

O

A

B

P

O

GEOMETRY

PACKAGE

552 Circle geometry (Chapter 27)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_27\552IGCSE01_27.CDR Monday, 27 October 2008 2:40:33 PM PETER