Circle Theorems (Including Proofs)

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Parts
A Reminder about parts of the Circle
Major Segment
diameter
chord
Minor Segment
Major Sector
Minor Sector
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Termgy
Introductory Terminology
Arc AB subtends angle x at the centre.
Arc AB subtends angle y at the circumference.
Chord AB also subtends angle x at the centre.
Chord AB also subtends angle y at the
circumference.
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Th1
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The angle subtended at the centre of a circle (by
an arc or chord) is twice the angle subtended at
the circumference by the same arc or chord.
(angle at centre)
Watch for this one later.
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42o (Angle at the centre).
70o(Angle at the centre)
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(180 2 x 42) 96o (Isos triangle/angle sum
triangle).
48o (Angle at the centre)
124o (Angle at the centre)
(180 124)/2 280 (Isos triangle/angle sum
triangle).
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Th2
This is just a special case of Theorem 1 and is
referred to as a theorem for convenience.
90o angle in a semi-circle
90o angle in a semi-circle
20o angle sum triangle
90o angle in a semi-circle
60o angle sum triangle
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Th3
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Angle x 30o
Angle x angle y 38o
Angle y 40o
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Th4
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If OT is a radius and AB is a tangent, find the
unknown angles, giving reasons for your answers.
180 (90 36) 54o Tan/rad and angle sum of
triangle.
90o angle in a semi-circle
60o angle sum triangle
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Th5
45o (Alt Seg)
60o (Alt Seg)
75o angle sum triangle
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Th6
The opposite angles of a cyclic quadrilateral are
supplementary. (They sum to 180o)
Angles p q 180o
Angles x w 180o
Angles y z 180o
Angles r s 180o
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180 85 95o (cyclic quad)
180 135 45o (straight line)
180 70 110o (cyclic quad)
180 110 70o (cyclic quad)
180 45 135o (cyclic quad)
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Th7
From any point outside a circle only two tangents
can be drawn and they are equal in length.
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PQ and PT are tangents to a circle with centre O.
Find the unknown angles giving reasons.
yo
Q
xo
O
90o (tan/rad)
98o
90o (tan/rad)
49o (angle at centre)
360o 278 82o (quadrilateral)
wo
zo
T
P
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PQ and PT are tangents to a circle with centre O.
Find the unknown angles giving reasons.
zo
Q
O
yo
90o (tan/rad)
xo
180 140 40o (angles sum tri)
50o (isos triangle)
50o (alt seg)
80o
wo
50o
T
P
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Th8
OS 5 cm (pythag triple 3,4,5)
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Angle SOT 22o (symmetry/congruenncy)
Angle x 180 112 68o (angle sum triangle)
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Mixed Q 1
65o (Alt seg)
130o (angle at centre)
25o (tan rad)
25o (isos triangle)
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Mixed Q 2
22o (cyclic quad)
68o (tan rad)
44o (isos triangle)
68o (alt seg)
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Geometric Proofs
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Proof 1/2
To prove that angle COB 2 x angle CAB

• Extend AO to D

• AO BO CO (radii of same circle)

• Triangle AOB is isosceles(base angles equal)

• Triangle AOC is isosceles(base angles equal)

• Angle AOB 180 - 2? (angle sum triangle)

• Angle AOC 180 - 2? (angle sum triangle)

• Angle COB 360 (AOB AOC)(lts at point)

• Angle COB 360 (180 - 2? 180 - 2?)

• Angle COB 2? 2? 2(? ?) 2 x lt CAB

QED
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Proof 3
To prove that angle CAB angle BDC

• With centre of circle O draw lines OB and OC.

• Angle COB 2 x angle CAB (Theorem 1).

• Angle COB 2 x angle BDC (Theorem 1).

• 2 x angle CAB 2 x angle BDC

• Angle CAB angle BDC

QED
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Proof 4
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To prove that the angle between a tangent and a
radius drawn to the point of contact is a right
angle.
To prove that OT is perpendicular to AB

• Assume that OT is not perpendicular to AB

• Then there must be a point, D say, on AB such
  that OD is perpendicular to AB.

O

• Since ODT is a right angle then angle OTD is
  acute (angle sum of a triangle).

• But the greater angle is opposite the greater
  side therefore OT is greater than OD.

B
T
A

• But OT OC (radii of the same circle) therefore
  OC is also greater than OD, the part greater than
  the whole which is impossible.

• Therefore OD is not perpendicular to AB.

• By a similar argument neither is any other
  straight line except OT.

• Therefore OT is perpendicular to AB.

Theorem 4
QED
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Proof 5
To prove that angle BTD angle TCD
2?
?

• With centre of circle O, draw straight lines OD
  and OT.

?

• Let angle DTB be denoted by ?.

• Then angle DTO 90 - ? (Theorem 4 tan/rad)

• Also angle TDO 90 - ? (Isos triangle)

• Therefore angle TOD 180 (90 - ? 90 - ?) 2?
  (angle sum triangle)

• Angle TCD ? (Theorem 1 angle at the centre)

• Angle BTD angle TCD

QED
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Proof 6
To prove that angles A C and B D 1800

• Draw straight lines AC and BD

• Chord DC subtends equal angles ? (same segment)

• Chord AD subtends equal angles ? (same segment)

• Chord AB subtends equal angles ? (same segment)

• Chord BC subtends equal angles ? (same segment)

• 2(? ? ? ?) 360o (Angle sum quadrilateral)

• ? ? ? ? 180o

QED
Angles A C and B D 1800
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Proof 7
To prove that AP BP.

• With centre of circle at O, draw straight lines
  OA and OB.

• OA OB (radii of the same circle)

• Angle PAO PBO 90o (tangent radius).

• Draw straight line OP.

• In triangles OBP and OAP, OA OB and OP is
  common to both.

• Triangles OBP and OAP are congruent (RHS)

• Therefore AP BP.

QED
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Proof 8
To prove that AB BC.

• From centre O draw straight lines OA and OC.

• In triangles OAB and OCB, OC OA (radii of same
  circle) and OB is common to both.

• Angle OBA angle OBC (angles on straight line)

• Triangles OAB and OCB are congruent (RHS)

QED

• Therefore AB BC

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Worksheet 1
Parts of the Circle
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Worksheet 2
Measure the angle subtended at the centre (y) and
the angle subtended at the circumference (x) in
each case and make a conjecture about their
relationship.
Th1
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Worksheet 3
To Prove that the angle subtended by an arc or
chord at the centre of a circle is twice the
angle subtended at the circumference by the same
arc or chord.
A
O
B
C
Theorem 1 and 2
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Worksheet 4
To Prove that angles subtended by an arc or chord
in the same segment are equal.
A
Theorem 3
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Worksheet 5
To prove that the angle between a tangent and a
radius drawn to the point of contact is a right
angle.
O
B
T
A
Theorem 4
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Worksheet 6
To prove that the angle between a tangent and a
chord through the point of contact is equal to
the angle subtended by the chord in the alternate
segment.
D
B
O
C
T
A
Theorem 5
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Worksheet 7
To prove that the opposite angles of a cyclic
quadrilateral are supplementary (Sum to 180o).
B
A
C
D
?
?
?
?
Theorem 6
Chi
delta
Alpha
Beta
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Worksheet 8
To prove that the two tangents drawn from a point
outside a circle are of equal length.
A
O
P
B
Theorem 7
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Worksheet 9
To prove that a line, drawn perpendicular to a
chord and passing through the centre of a circle,
bisects the chord.
O
A
C
B
Theorem 8