31

.1

Isosceles triangle in a circle

31.2

Tangents to a circle

© The authors 2010. E

dexcel GCSE Mathematics A Linear. This document may have been altered from the original

SPECIFICATION REFERE

NCES

GCSE 2010

GM j Understand and construct geometrical proofs using circle theorems

FS Process skills

Select the mathematical information to use

Use appropriate mathematical procedures

Choose appropriate language… to communicate resul

ts and solutions

FS Performance

Level 1 Select mathematics in an organised way to find solutions

Resources

Pairs of compasses, rulers

Concepts and skills

Understand and use the fact that the tangent at any point on a circle is perpendicular to the radius

at that point.

Understand and use the fact that tangents from an external point are equal in length.

Find missing angles on diagrams.

Give reasons for angle calculations involving the use of tangent theorems.

Functional skills

L1 Construct geometric diagr

ams, models and shapes.

Prior key knowledge, skills and concepts

Students should already know that the angles in a triangle add up to 180° and the angles in a

quadrilateral to 360°, and be able to name the parts of a circle.

Starter

Draw a circle.

Draw t

wo radii and then draw a line between the two points where the radii meet the circumference.

What sort of triangle is this?

(Isosceles)

Draw two tangents from the points where the two radii meet the circumference, so that the tangents

meet at a point outsi

de the circle.

Measure the length between each radius and the tangent it touches.

(This will be a right angle.)

Measure the length of each tangent from the point outside the circle to where the tangent touches

the circle.

(The lengths will be the same.)

31

.1

Isosceles triangle in a circle

31.2

Tangents to a circle

© The authors 2010. E

dexcel GCSE Mathematics A Linear. This document may have been altered from the original

Main teaching and learning

Tell students that they are going to learn about an isosceles triangle in a circle, as the triangle is

made up partly of two radii, and about the properties of tangents to a circle.

Introduce students to the idea of a formal

geometric proof, stressing the fact that they need to know

how to show each stage of the working, giving a formal reason for each stage of the proof.

Go through

some examples

in detail, explaining to students that they need to be able to do these

proofs th

emselves.

Tell students that every time they see a geometric problem with a tangent, that they need to use the

fact that the angle between a tangent and a radius is 90°, or that two tangents to a circle are equal

in length.

Common misconceptions

Students o

ften state that the angle between the tangent and the

circle

is 90°. This is insufficient in

an examination. They need to state that the angle between the tangent and the

radius

is 90°.

Remind students they have to explain themselves fully using formal ge

ometric rules at each stage

of the proof, rather than just calculations.

Plenary

Put students in pairs where one student makes up a circle problem using tangents and the other

student has to solve the problem. They then swap roles.

31

.

3 Circle theorems

© The authors 2010. E

dexcel GCSE Mathematics A Linear. This document may have been altered from the original

SPECIFICATION REFERE

NCES

GCSE 2010

GM j Understand and construct geometrical proofs using circle theorems

FS Process skills

Select the mathematical information to use

Use appropriate mathematical procedures

Choose appropriate language…to communicate results and solutions

FS Perfo

rmance

Level 1 Select mathematics in an organised way to find solutions

RESOURCES

Resources

Pairs of compasses, rulers

Links

http://lgfl.skoool.co.uk/keystage4.aspx?id=317

Concepts and skills

Find missing angles on diagrams.

Prove and use the facts that

the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point

on the circumference

the angle in a semicircle is a right angle

the perpendicular from the centre of a circle to a chord bisects the chord.

Functional skills

L1 Construct geometric diagrams, models and shapes.

Prior key knowledge, skills and concepts

Students should already know how to state formally that

the angles in a triangle add up to 180°

the angles on a straight line add up to 180°

the angles opposite

the equal sides of an isosceles triangle are equal

the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

31

.

3 Circle theorems

© The authors 2010. E

dexcel GCSE Mathematics A Linear. This document may have been altered from the original

Starter

Put on the board some diagrams, with unknown angle values that can be found using what students

h

ave already learned about isosceles triangles in a circle and tangents to a circle, e.g.

Work out the missing angles and give a formal reason for your answers.

(Answers:

a

= 35º,

b

= 27º)

Main teaching and learning

Tell students they are goin

g to learn some circle theorems, explaining that they need to be able to

formally prove each theorem.

Go through the three theorems in detail and then

go through some exercises.

Common misconceptions

Tell students that when they draw circles using their pa

ir of compasses, it is best not to hold on to

the pencil end as this tends to make the end move, resulting in an inaccurately drawn circle.

Plenary

Give students copies of some problems involving circle theorems and supply a partial explanation.

Ask them t

o suggest better wording.

31

.

4 More circle theorems

© The authors 2010. E

dexcel GCSE Mathematics A Linear. This document may have been altered from the original

SPECIFICATION

GCSE 2010

GM j Understand and construct geometrical proofs using circle theorems

FS Process skills

Select the mathematical information to use

Use appropriate mathematical procedures

Choose appropriate language … to

communicate results and solutions

FS Performance

Level 1 Select mathematics in an organised way to find solutions

Key words

cyclic quadrilateral, proof, prove

Resources

Pairs of compasses, rulers

Concepts and skills

Find missing angles on diagrams.

Prov

e and use the facts that

angles in the same segment are equal

opposite angles of a cyclic quadrilateral sum to 180º

alternate segment theorem.

Functional skills

L1 Construct geometric diagrams, models and shapes.

Prior key knowledge, skills and concepts

St

udents should already know how to state formally that

the angles in a triangle add up to 180°

the angles on a straight line add up to 180°

the angles opposite the equal sides of an isosceles triangle are equal

the exterior angle of a triangle is equal to t

he sum of the two interior opposite angles.

31

.

4 More circle theorems

© The authors 2010. E

dexcel GCSE Mathematics A Linear. This document may have been altered from the original

Starter

Put on the board some diagrams, with unknown angle values that can be found using the circle

theorems already covered.

Work out the missing angles and give a formal reason for your answers.

e.

g.

(

a

= 53º, because the

(

b

= 34º, because the angle

(

c

= 57º, because the angle in

perpendicular from the

at the centre of the circle

a semicircle is a right angle.)

centre of a circle to a

is twice the angle at the

chord bisects

the chord.)

circumference.)

Main teaching and learning

Tell students they are going to learn some more circle theorems, explaining that they need to be

able to formally prove each theorem.

Go throu

gh the three theorems in detail

and then

go through some e

xercises.

Plenary

Give students copies of some problems involving circle theorems and supply a partial explanation.

Ask them to suggest better wording.

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