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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