Grade 11 Euclidean Geometry 2014
GRADE 11 EUCLIDEAN GEOMETRY
4.
CIRCLES
4.1
Arc
Chord
Radius
Diameter
Segment
Tangent
TERMINOLOGY
An arc is a part of the circumference of a circle
A chord is a straight line joining the ends of an arc.
A radius is any straight line from the centre of the circle to a point on the
circumference
A diameter is a special chord that passes through the centre of the circle. A
Diameter is the length of a straight line segment from one point on the
circumference to another point on the circumference, that passes through the
centre of the circle.
A segment is the part of the circle that is cut off by a chord. A chord divides a
circle into two segments
A tangent is a line that makes contact with a circle at one point on the
circumference (AB is a tangent to the circle at point P).
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Grade 11 Euclidean Geometry 2014
4.2 SUMMARY OF THEOREMS
4.2.1
Definitions
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Grade 11 Euclidean Geometry 2014
4.2.2
Chords and Midpoints
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Grade 11 Euclidean Geometry 2014
4.2.3 Angles in circles
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Grade 11 Euclidean Geometry 2014
4.2.4
Cyclic Quadrilaterals
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Grade 11 Euclidean Geometry 2014
4.2.4
Tangents
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Grade 11 Euclidean Geometry 2014
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Grade 11 Euclidean Geometry 2014
4.3 PROOF OF THEOREMS
All SEVEN theorems listed in the CAPS document must be proved. However, there are four
theorems whose proofs are examinable (according to the Examination Guidelines 2014) in
grade 12. In this guide, only FOUR examinable theorems are proved. These four theorems
are written in bold.
1. The line drawn from the centre of a circle perpendicular to the chord bisects the
chord.
2. The perpendicular bisector of a chord passes through the centre of the circle.
3. The angle subtended by an arc at the centre of a circle is double the angle subtended
by the same arc at the circle (on the same side of the arc as the centre).
4. Angles subtended by an arc or chord of the circle on the same side of the chord are equal.
5. The opposite angles of a cyclic quadrilateral are supplementary.
6. Two tangents drawn to a circle from the same point outside the circle are equal in length
(If two tangents to a circle are drawn from a point outside the circle, the distances
between this point and the points of contact are equal).
7. The angle between the tangent of a circle and the chord drawn from the point of
contact is equal to the angle in the alternate segment.
The above theorems and their converses, where they exist, are used to prove riders.
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Grade 11 Euclidean Geometry 2014
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Grade 11 Euclidean Geometry 2014
OR
Theorem 1
The line drawn from the centre of a circle, perpendicular to a chord, bisects the chord.
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Grade 11 Euclidean Geometry 2014
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Grade 11 Euclidean Geometry 2014
OR
Theorem 4
The angle subtended by an arc at the centre of a circle is double the size of the angle
subtended by the same arc at the circumference of the circle.
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Grade 11 Euclidean Geometry 2014
Theorem 7
The opposite angles of a cyclic quadrilateral are supplementary.
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Grade 11 Euclidean Geometry 2014
Theorem 10
The angle between a tangent and a chord, drawn at the point of contact, is equal to the angle
which the chord subtends in the alternate segment.
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Grade 11 Euclidean Geometry 2014
4.4
ACTIVITIES
GEOMETRY 1
1. The sketch alongside shows chord BD cutting AE at C. A is the
centre of the circle and AE ⊥ BD. If EC = 3cm and BD = 14cm,
calculate the area of the circle.
E
C
B
D
A
2. The sketch shows circle centre O with OC ‖ AB . OCˆ B = 76º and  = x.
Calculate x.
O
76
A
15
x
B
o
C
Grade 11 Euclidean Geometry 2014
CONDITIONS FOR QUADRILATERAL TO BE CYCLIC
If
OR
opp. int.
angles
suppl.
Then PQRS is a
cyclic quad.
If
OR
Then PQRS is a
cyclic quad.
If
Then PQRS is a
cyclic quad.
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Angles in
the same
seg.
ext. angle
equal to int.
opp. angle
Grade 11 Euclidean Geometry 2014
3. The diagram shows circles with centres Q and O, and MTˆR = 400.
MT and RT are not necessarily tangents to the smaller circle.
M
P
1
2
3
4
1
Q
1
2
1
2
3
R
Determine:
3.1
Q̂2
3.2
Ô1
3.3
PMˆ O
3.4
P̂
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O
o
40
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Grade 11 Euclidean Geometry 2014
4.
In the accompanying figure, AB is a diameter of the circle with centre O. DC is a
tangent to the circle at point C. Chord AC is drawn. D is a point on the tangent DC so
that Aˆ1 = Aˆ 2 .
A
D
2
1
O
2
1
2
1
C
B
E
Prove that:
4.1 AD ‖ OC
4.2 ADˆ C = 90
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Grade 11 Euclidean Geometry 2014
5.
In the figure PS is a diameter of the circle with centre T. BQ is a tangent to the circle
and TR is perpendicular to QS. RTˆS = x.
B
1
2
Q
3 4
1
R
2
P
1
2
x
1
2
T
S
5.1 Prove that TR ‖ PQ.
5.2 Determine, with reasons, other four angles each equal to x.
5.3 Prove that TQRS is a cyclic quadrilateral.
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Grade 11 Euclidean Geometry 2014
6.
In the figure below, diagonals AC and BD of cyclic quadrilateral ABCD
intersect at P such that AP = PB. FPG is a tangent to circle
F
D
2
A
1
2
1
2 1
P
3
4
1
2
2
B
1
C
G
ABP.
Prove that:
6.1 FG ‖ DC
6.2
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Grade 11 Euclidean Geometry 2014
7. The sketch below shows circles BKAC and KMTB intersecting at K and B, and
ABˆ T = 90 . AB and BT are not diameters, BT is not a tangent to the smaller circle, and
AB is not a tangent to the larger circle.
B
3
1 2
C
1
A
2
2
3
1
3
K
2
4
1
2
1
M
2
1
S
7.1 Prove that SABT is a cyclic quadrilateral.
7.2 Express in terms of .
7.3 Prove that
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Grade 11 Euclidean Geometry 2014
SOLUTIONS
GEOMETRY 1
E
1. BC = CD = 7cm ….. AC ⊥ BD
Let AC = x
C
B
Then the radius r = x + 3
D
AD² = AC² + CD² … Pythag
Thus (x + 3)² = x² + 7²
A
 x² + 6x + 9 = x² + 49
 6x = 40
 x = 6⅔
 r = 9⅔
 Area =  (9⅔) ²
= 293, 56 cm²
2. ABˆ C = 104º …. Co-int ’s;
OC ‖ AB.
 Reflex AOˆ C = 2B
= 208º
 Obtuse AOˆ C = 152º
O
76
 x = 28º … co-int ’s;
OC ‖ AB.
A
x
B
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o
C
Grade 11 Euclidean Geometry 2014
3.
M
P
1
2
3
4
1
Q
1
2
1
R
3.1 Q̂ 2 = 140º ….. opposite T̂ in cyclic quad MQRT
3.2 Ô1 = 80º ….. 2× T̂ on the circumference
3.3 PMˆ R = 90º …..  in a semi-circle
M 3 = 50º …..  sum of isosceles ∆ OMR
 PMˆ O = 1400
3.4 P̂ = 70º ….. Q̂ 2 is the exterior  of isosceles ∆ QMP
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2
3
O
o
40
T
Grade 11 Euclidean Geometry 2014
A
4.
D
2
1
O
2
1
2
1
C
B
E
4.1
C2= A
1
….. OA = OC (radii)
 C 2 = A 2 ….. A 1= A
2
(given)
 AD ‖ OC ….. alternate angles equal
4.2 OĈE = 90º ….. tangent CE ⊥ radius OC
 A Dˆ C = 90 ….. corresponding angles; AD ‖ OC
5.
B
5.1 Q 2 + 3 = 90º ….  in semi-circle
 TR ‖ PQ …. corresponding ’s equal
5.2 T 2 = x …. TR is a line of symmetry of isos  TQS
1
2
 P = x …. ½ T at the centre
Q
3 4
1
R
2
 Q 2 = x …. PT = TQ (radii)
P
Q 4 = P = x …. tan BQR ; chord QS
1
T
24
2
x
1
2
S
Grade 11 Euclidean Geometry 2014
5.3 Q 4 = RTˆS = x. Thus TQRS is cyclic …. chord RS
6.
F
D
2
A
1
2
1
2 1
P
3
4
1
2
2
B
1
C
G
6.1)  P4 = A1 ..... tang PG; chord PB
But P4 = P1 ..... vert opp angles and A1 = D1 ... chord BC
 P1 = D1
 FG ‖ DC … alt’s equal
6.2 P4 = A1 ..... tang PG; chord PB
P2 = B1 …. tang FP; chord AP
But A1 = B1 …. equal chords AP and BP
 P4 = P2 = P1
 FP bisects AP̂D
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Grade 11 Euclidean Geometry 2014
7.
B
3
1 2
C
1
A
2
2
3
1
3
K
2
4
1
2
1
M
2
1
S
7.1
M1 = B 3 ….. exterior  of cyclic quad BKMT
Let M 1 = B 3 = x
 ABˆ K = 90º – x = Ĉ ….. ’s in the same segment
 ASˆM = 90º …..  sum ∆ CSM
 BAST is cyclic ….. ABˆ T + ASˆM = 180
7.2 Let T2 = y
 S 2 = y ….. same chord AB
 S1 = 90º – S 2 = 90º – T2
7.3
M 1 = K 4 + T 1 ….. ext  of ∆
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Grade 11 Euclidean Geometry 2014
But M 1 = B 3 , K 4 = K 1 (vert opp ’s) and T 1 = B 1 (chord AS)
 Bˆ3 = Kˆ 1 + Bˆ1
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Grade 11 Euclidean Geometry 2014
MORE ACTIVITIES
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Grade 11 Euclidean Geometry 2014
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Grade 11 Euclidean Geometry 2014
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