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S5 Maths: Angles in Circles: Paper 3 (challenge)
S5 Maths: Angles in Circles: Paper 3 (challenge)
Ten items mixing unknowns, arc ratios, chord lengths and the reflex angle trap.
Direct answers Items 1-3: 74, 126, 23 degrees. Items 4-6: 80 degrees, x = 15 with 55 degrees, 150 degrees. Items 7-9: x = 20 with 65 degrees, x = 35 with 115 degrees, 10 cm. Item 10: 50 degrees.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S5 Maths: Angles in Circles |
| Level | Challenge |
02 Questions (10 items)
- FoundationThe central angle AOB = 148 degrees. Find the inscribed angle on the same arc.
- FoundationAn inscribed angle is 63 degrees. Find the central angle standing on the same arc.
- FoundationAB is a diameter and the angle ABC = 67 degrees. Find the angle BAC.
- IntermediateThree angles of the cyclic quadrilateral ABCD are 85, 95 and 100 degrees. Find the fourth angle.
- IntermediateIn the same segment the angle ACB = 4x – 5 and the angle ADB = 2x + 25. Find x and the angle.
- IntermediateThe arcs AB, BC and CA are in the ratio 5 : 4 : 3. Find the largest central angle.
- ChallengeThe central angle AOB = 7x – 10 and the inscribed angle ACB = 3x + 5 on the same arc. Find x and the inscribed angle.
- ChallengeIn the cyclic quadrilateral ABCD the angle at A is 3x + 10 and the angle at C is x + 30. Find x and the angle at A.
- ChallengeA circle has radius 10 cm and the central angle AOB = 60 degrees. Find the length of the chord AB.
- Error redoA student sees a central angle of 100 degrees on the arc AB and uses 360 – 100 = 260 as the inscribed angle. What is the correct value?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 74 degrees | Half of the central angle is 148 / 2 = 74 degrees. |
| 2 | 126 degrees | Double it: 63 x 2 = 126 degrees. |
| 3 | 23 degrees | The angle ACB is 90 degrees, so the angle BAC is 90 – 67 = 23 degrees. |
| 4 | 80 degrees | The angles of a quadrilateral add to 360 degrees, so 360 – 280 = 80 degrees. |
| 5 | x = 15 and the angle is 55 degrees | Equal angles in the same segment give 4x – 5 = 2x + 25, so x = 15 and the angle is 55 degrees. |
| 6 | 150 degrees | The largest share is 5 of 12 parts, so 360 x 5 / 12 = 150 degrees. |
| 7 | x = 20 and the inscribed angle is 65 degrees | From 7x – 10 = 2(3x + 5) we get x = 20 and the inscribed angle is 65 degrees. |
| 8 | x = 35 and the angle at A is 115 degrees | Opposite angles add to 180 degrees, so 4x + 40 = 180 gives x = 35 and the angle at A is 115 degrees. |
| 9 | 10 cm | With two equal radii and a 60 degree angle the triangle OAB is equilateral, so AB = 10 cm. |
| 10 | 50 degrees | The inscribed angle is half the central angle, so 100 / 2 = 50 degrees; 260 degrees is the reflex angle. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
After solving, substitute x back into both angles and check that the central angle really is double.Tutor Registration →
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.