myUASARevision for UASA
More
My progress
Mathematics questions, chapter 6
Nama / NameKelas / ClassTarikh / Date

Angles and Tangents of Circles

40 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working and give a reason for each step in Section B. Diagrams are not drawn to scale unless the question says so.

Section A: objective questions

12 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

Diagram 1 shows a circle with centre O. Which angle is the angle at the circumference subtended by the minor arc PQ?
Circle with centre O, chord PQ near the bottom and point R on the major arc; lines join R to P and Q, and O to P and Q
Diagram 1
Show answerHide answer
B
The angle at the circumference has its vertex on the circle (R) and its arms end at P and Q, the two ends of the arc. ∠POQ has its vertex at the centre, so it is the central angle. ∠OPQ and ∠RPQ have their vertex at P.
For the teacher: Remembering, low difficulty, standard 6.1.1
Textbook p. 130
2

Question 2

In Diagram 2, P, Q, R and S lie on a circle. ∠PRQ = 38°. Find the value of x.
Circle with points P and Q at the bottom and R and S above; R and S each joined to P and Q; angle PRQ is 38 degrees and angle PSQ is x
Diagram 2
Show answerHide answer
C
x and 38° are both angles at the circumference subtended by arc PQ, so they are equal: x = 38°. 76° is the central angle on arc PQ and 19° is half of 38°.
For the teacher: Understanding, low difficulty, standard 6.1.1
Textbook p. 132
3

Question 3

In Diagram 3, arc PQ = 4 cm and arc RS = 12 cm. Find the value of x.
Circle with arc PQ of 4 cm subtending 15 degrees at T and arc RS of 12 cm subtending x at U
Diagram 3
Show answerHide answer
A
The angle at the circumference is proportional to the arc length. Arc RS is 12 ÷ 4 = 3 times arc PQ, so x = 3 × 15° = 45°. 60° comes from multiplying by 4 instead of 3; 23° from adding (15 + 12 − 4).
For the teacher: Applying, low difficulty, standard 6.1.1
Textbook p. 136
4

Question 4

Diagram 4 shows a circle with centre O. ∠POQ = 112° and R lies on the minor arc PQ. Find the value of x.
Circle with centre O; P at the upper left and Q at the lower left; R on the minor arc between them at the far left; angle POQ is 112 degrees and angle PRQ is x
Diagram 4
Show answerHide answer
D
R is on the minor arc, so ∠PRQ is subtended by the major arc, whose central angle is 360° − 112° = 248°. x = ½ × 248° = 124°. 56° would be correct only if R were on the major arc.
For the teacher: Applying, medium difficulty, standard 6.1.2
Textbook p. 139
5

Question 5

In Diagram 5, PQ is a diameter of the circle with centre O. ∠RPQ = 35°. Find the value of x.
Circle with horizontal diameter PQ through centre O and R on the circumference above; angle RPQ is 35 degrees and angle PQR is x
Diagram 5
Show answerHide answer
B
The angle in a semicircle ∠PRQ = 90°. In triangle PQR, x = 180° − 90° − 35° = 55°. 90° is ∠PRQ and 125° is 180° − 55°.
For the teacher: Understanding, low difficulty, standard 6.1.1
Textbook p. 142
6

Question 6

Which statement describes a cyclic quadrilateral?
Show answerHide answer
A
A cyclic quadrilateral has all four vertices on the circumference. A circle inside that touches the four sides is a different idea and does not make the quadrilateral cyclic.
For the teacher: Remembering, low difficulty, standard 6.2.1
Textbook p. 144
7

Question 7

In Diagram 6, PQRS is a cyclic quadrilateral and QRT is a straight line. ∠SRT = 82°. Find ∠SPQ.
Cyclic quadrilateral PQRS with side QR extended past R to T; the exterior angle SRT is 82 degrees
Diagram 6
Show answerHide answer
D
The exterior angle SRT equals the opposite interior angle SPQ, so ∠SPQ = 82°. Check: ∠QRS = 180° − 82° = 98°, and ∠SPQ + ∠QRS = 180°. 98° is the interior angle at R, not at P.
For the teacher: Applying, low difficulty, standard 6.2.2
Textbook p. 147
8

Question 8

Diagram 7 shows a circle with centre O and four straight lines. Which line is a tangent to the circle?
Circle with centre O. Line AB cuts through the circle; line CD touches the circle at one point; line EF is outside the circle; line GH passes through the centre O
Diagram 7
Show answerHide answer
C
A tangent touches the circle at exactly one point. CD touches once. AB and GH cut the circle at two points, and EF does not touch it at all.
For the teacher: Understanding, low difficulty, standard 6.3.1
Textbook p. 150
9

Question 9

In Diagram 8, O is the centre of the circle and AB is a tangent that touches the circle at B. ∠AOB = 64°. Find the value of x.
Circle with centre O; tangent line through A and B touching the circle at B directly below O; angle AOB is 64 degrees and angle OAB is x
Diagram 8
Show answerHide answer
B
The radius OB is perpendicular to the tangent, so ∠OBA = 90°. x = 180° − 90° − 64° = 26°.
For the teacher: Applying, low difficulty, standard 6.3.2
Textbook p. 152
10

Question 10

In Diagram 9, PQR is a tangent to the circle at Q. ∠PQS = 60° and ∠SQT = 72°. Find the value of x.
Tangent PQR touching a circle at Q; chords QS and QT; angle PQS is 60 degrees, angle SQT is 72 degrees and angle QST is x
Diagram 9
Show answerHide answer
A
On the straight line PQR: ∠RQT = 180° − 60° − 72° = 48°. x = ∠QST is in the alternate segment of chord QT, so x = ∠RQT = 48°. 60° is ∠PQS, which matches the angle at T, not at S.
For the teacher: Analysing, medium difficulty, standard 6.3.4
Textbook p. 155
11

Question 11

Diagram 10 shows a cyclic quadrilateral PQRS. ∠PQR = (2x + 10)° and ∠PSR = (3x − 20)°. Find the value of x.
Cyclic quadrilateral PQRS with angle PQR labelled (2x + 10) degrees and angle PSR labelled (3x − 20) degrees
Diagram 10
Show answerHide answer
C
∠PQR and ∠PSR are opposite angles, so (2x + 10) + (3x − 20) = 180. 5x − 10 = 180, 5x = 190, x = 38. Check: 86° + 94° = 180°.
For the teacher: Applying, medium difficulty, standard 6.2.3
Textbook p. 148
12

Question 12

In Diagram 11, PA and PB are tangents to the circle with centre O. ∠APB = 48°. Find the value of x.
Tangents PA and PB to a circle centre O, with chord AB drawn; angle APB is 48 degrees and angle OAB is x
Diagram 11
Show answerHide answer
D
PA = PB, so triangle PAB is isosceles and ∠PAB = (180° − 48°) ÷ 2 = 66°. OA is perpendicular to the tangent, so ∠OAP = 90° and x = 90° − 66° = 24°. Check: ∠AOB = 360° − 90° − 90° − 48° = 132°, and (180° − 132°) ÷ 2 = 24°.
For the teacher: Analysing, high difficulty, standard 6.3.3
Textbook p. 154

Section B: structured questions

28 marks
Answer all questions. Show your working and give a reason for each step.
13

Question 13

Diagram 12 shows a circle with centre O. R lies on the major arc PQ and S lies on the minor arc PQ. ∠POQ = 84°.
Circle with centre O, angle POQ 84 degrees at the bottom; R on the major arc at the top and S on the minor arc below O; R and S joined to P and Q
Diagram 12
[1 mark] (a)Find ∠PRQ. Give a reason.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Find ∠PSQ. Give a reason.
Show answerHide answer
Check: ∠PRQ + ∠PSQ = 42° + 138° = 180°.
Tick each point you got.
For the teacher: Applying, low difficulty, standard 6.1.2
Textbook p. 139
14

Question 14

Diagram 13 shows a circle. Arc PQ = 5 cm and arc QS = 15 cm. ∠PRQ = 24°.
Circle with arc PQ of 5 cm subtending 24 degrees at R and arc QS of 15 cm subtending x at T
Diagram 13
[1 mark] (a)Find the angle at the centre subtended by arc PQ.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Find the value of x.
Show answerHide answer
Tick each point you got.
[1 mark] (c)Calculate the circumference of the circle.
Show answerHide answer
360° ÷ 48° = 7.5, so the whole circumference is 7.5 pieces of 5 cm.
Tick each point you got.
For the teacher: Applying, low difficulty, standard 6.1.2
Textbook p. 136
15

Question 15

Diagram 15 shows a cyclic quadrilateral ABCD in a circle with centre O. ∠AOC = 150° and ∠CAD = 40°.
Circle with centre O; cyclic quadrilateral ABCD with B at the top and D at the bottom; angle AOC is 150 degrees and angle CAD is 40 degrees
Diagram 15
[1 mark] (a)Find ∠ABC.
Show answerHide answer
Tick each point you got.
[1 mark] (b)Find ∠ADC.
Show answerHide answer
Tick each point you got.
[1 mark] (c)Find ∠ACD.
Show answerHide answer
Tick each point you got.
[1 mark] (d)Find ∠ABD. Give a reason.
Show answerHide answer
Tick each point you got.
For the teacher: Analysing, low difficulty, standard 6.2.3
Textbook p. 148
16

Question 16

Diagram 16 shows a circle with centre O and radius 5 cm. PA and PB are tangents to the circle at A and B. OP = 13 cm.
Circle centre O with radius OA of 5 cm; tangents PA and PB from P with OP 13 cm; right angles at A and B
Diagram 16
[1 mark] (a)Calculate the length of PA.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Calculate ∠APO, correct to one decimal place.
Show answerHide answer
Tick each point you got.
[1 mark] (c)Hence find ∠APB.
Show answerHide answer
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 6.3.3
Textbook p. 154
17

Question 17

Diagram 17 shows a circle. PAQ is a tangent to the circle at A. AB = AC and ∠PAB = 66°.
Horizontal tangent PAQ touching a circle at A at the top; triangle ABC inscribed with B lower left and C lower right; angle PAB is 66 degrees; AB and AC are marked equal
Diagram 17
[1 mark] (a)Find ∠ACB. Give a reason.
Show answerHide answer
Tick each point you got.
[1 mark] (b)Find ∠ABC.
Show answerHide answer
Tick each point you got.
[1 mark] (c)Find ∠BAC.
Show answerHide answer
Tick each point you got.
[1 mark] (d)Show that PQ is parallel to BC.
Show answerHide answer
Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 6.3.4
Textbook p. 155
18

Question 18

Diagram 18 shows two circles with centres A and B. Their radii are 8 cm and 3 cm. AB = 13 cm. PQ is a common tangent that touches the circles at P and Q.
Two separate circles with centres A and B, radii 8 cm and 3 cm, AB 13 cm; the common tangent PQ touches the large circle at P and the small circle at Q
Diagram 18
[1 mark] (a)State the number of common tangents the two circles have.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Calculate the length of PQ.
Show answerHide answer
Tick each point you got.
[1 mark] (c)The small circle is moved along AB until it touches the large circle from outside. State the number of common tangents now.
Show answerHide answer
Tick each point you got.
For the teacher: Applying, low difficulty, standard 6.3.5
Textbook p. 158
19

Question 19

In Diagram 19, O is the centre of the circle and ∠AOC = 140°. Aini says, "∠ABC = 70° because the angle at the circumference is half of the central angle."
Circle with centre O; A at the left and C at the right; B on the circumference below, on the minor arc AC; angle AOC is 140 degrees
Diagram 19
[1 mark] (a)Is Aini correct for the point B in Diagram 19?
Show answerHide answer
No
[2 marks] (b)Explain Aini's mistake and find the correct value of ∠ABC.
Show answerHide answer
Aini is right only when B is on the major arc AC. Check with a cyclic quadrilateral: 70° + 110° = 180°.
Tick each point you got.
For the teacher: Evaluating, high difficulty, standard 6.1.2
Textbook p. 139
20

Question 20

O is the centre of a circle. Chord PQ subtends ∠POQ = 100° at the centre. Sketch the circle and mark a point R on the circumference so that ∠PRQ = 50°, and a point S on the circumference so that ∠PSQ = 130°.
[2 marks] (a)State on which arc (major or minor) PQ each point lies and give a reason.
Show answerHide answer
R on the major arc PQ (50° = ½ of 100°); S on the minor arc PQ (130° = ½ of 260°). Any point on the correct arc will do.
Tick each point you got.
For the teacher: Creating, medium difficulty, standard 6.1.2
Textbook p. 139