Circle Geometry Riders: Proofs, Converses & Method (Grade 11)

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Grade 11 · Euclidean Geometry

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 31

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
The line drawn from the centre of a circle perpendicular to a chord …
(1)
A)is equal to the chord
B)is a tangent
C)halves the radius
D)bisects the chord✓
Answer: D — line from centre ⊥ chord.
- A — its length depends on the radius
- B — a tangent touches the circle at one point only
- C — nothing forces it to be half the radius
1.2
O is the centre and AÔB = 140°. The angle at the circumference subtended by the same arc AB is …
(1)
A)140°
B)280°
C)70°✓
D)40°
Answer: C — ∠ at centre = 2 × ∠ at circumference, so 140° ÷ 2 = 70°.
- A — the angle at the circumference is HALF the angle at the centre
- B — doubled instead of halving
- D — that is 180° − 140°
1.3
In cyclic quadrilateral PQRS, P̂ = 105°. Then R̂ = …
(1)
A)105°
B)75°✓
C)255°
D)15°
Answer: B — opp ∠s of cyclic quad: 180° − 105° = 75°.
- A — opposite angles of a cyclic quad are supplementary, not equal
- C — subtracted from 360° instead of 180°
- D — subtracted from 90°
1.4
To PROVE that ABCD is cyclic you may use …
(1)
A)a pair of opposite angles that add up to 180°✓
B)opp ∠s of a cyclic quad
C)the angle at the centre theorem
D)a pair of equal chords
Answer: A — The CONVERSE: if opposite angles are supplementary, the quadrilateral is cyclic.
- B — that reason assumes the quadrilateral is already cyclic
- C — a proof of cyclicity has no centre to work from
- D — equal chords do not make a quadrilateral cyclic
1.5
A chord of 24 cm lies in a circle of radius 13 cm. Its distance from the centre is …
(1)
A)12 cm
B)11 cm
C)√313 cm
D)5 cm✓
Answer: D — The perpendicular bisects the chord: √(13² − 12²) = √25 = 5 cm.
- A — that is half the chord
- B — subtracted 13 − 2 instead of using Pythagoras
- C — ADDED the squares instead of subtracting
1.6
The angle between a tangent and a chord at the point of contact equals …
(1)
A)the angle at the centre
B)90°
C)the angle in the alternate segment✓
D)the angle in the same segment
Answer: C — tan–chord theorem.
- A — the angle at the centre is twice the angle in the alternate segment
- B — that is the angle between a tangent and a RADIUS
- D — it is the OTHER segment
1.7
PA and PB are tangents from P to a circle with centre O. If AP̂B = 50°, then AÔB = …
(1)
A)50°
B)130°✓
C)100°
D)25°
Answer: B — OÂP = OB̂P = 90° (tan ⊥ radius), so AÔB = 360° − 90° − 90° − 50° = 130°.
- A — the angles of OAPB add up to 360°, not equal in pairs
- C — doubled the 50° as if it were an angle at the circumference
- D — halved the 50°
1.8
The exterior angle of a cyclic quadrilateral equals …
(1)
A)the interior opposite angle✓
B)the adjacent interior angle
C)90°
D)twice the opposite angle
Answer: A — ext ∠ of cyclic quad.
- B — those two are supplementary
- C — no right angle is involved
- D — that describes the angle at the centre
1.9
Angles at the circumference subtended by the same arc are …
(1)
A)supplementary
B)complementary
C)in the ratio 1 : 2
D)equal✓
Answer: D — ∠s in the same segment.
- A — that is opposite angles of a cyclic quad
- B — no 90° relationship is involved
- C — that compares the circumference with the centre
1.10
The FIRST step in proving that the perpendicular from the centre bisects a chord is to …
(1)
A)measure the two halves of the chord
B)draw a tangent
C)join the centre to both ends of the chord✓
D)assume the chord is bisected
Answer: C — The construction (two radii) sets up the congruent triangles.
- A — measuring is never a proof
- B — no tangent is needed
- D — that assumes the conclusion

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[13 MARKS]

In the diagram below, O is the centre of the circle. A, B, C and D lie on the circle, BD is a diameter and DB̂C = 32°. The diagram is not drawn to scale.

2.1
Determine, with reasons, the sizes of BĈD and BD̂C.
(4)
BĈD = 90° (1) (∠ in semicircle) (1)
BD̂C = 180° − 90° − 32° = 58° (1) (∠s of △) (1)
2.2
Determine BÂC, with a reason.
(2)
BÂC = BD̂C = 58° (1) (∠s in same segment) (1)
2.3
Determine BÔC, with a reason.
(3)
BÔC = 2 × BD̂C (1) = 116° (1) (∠ at centre = 2 × ∠ at circumference) (1)
2.4
Explain why △OBC is isosceles, and hence calculate OB̂C.
(4)
OB = OC (radii) (1), so △OBC is isosceles (1)
OB̂C = (180° − 116°)/2 (1) = 32° (1) — as it must be, because O lies on BD

Question 3

[8 MARKS]

In the diagram below, TA is a tangent to the circle at A. B and C lie on the circle, TÂB = 55° and AB̂C = 70°.

3.1
Determine AĈB, with a reason.
(2)
AĈB = TÂB = 55° (1) (tan–chord theorem) (1)
3.2
Determine BÂC, with a reason.
(3)
BÂC = 180° − 70° − 55° (1) = 55° (1) (∠s of △) (1)
3.3
Prove that AB = BC.
(3)
BÂC = AĈB = 55° (1)
sides opposite equal angles are equal (1)
∴ AB = BC (1)

TOTAL: 31 marks

This question paper consists of 3 questions.