Short Introduction
Exercise 5.6 introduces one of the most beautiful properties of circles:
The angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the remaining part of the circle.
Students also learn that all angles standing on the same chord are equal and use these results to solve geometrical problems.
Quick Information Box
| Particular | Details |
|---|---|
| Class | 9 |
| Subject | Mathematics |
| Chapter | 5 – I’m Up and Down, and Round and Round |
| Exercise | 5.6 |
| Main Topic | Angles Subtended by Arcs |
| Difficulty Level | Moderate |
| Important Theorem | Angle at Centre = 2 × Angle at Circumference |
Concepts Used (Topics Covered)
- Arc and Chord
- Angle Subtended by an Arc
- Central Angle
- Angle in the Same Segment
- Diameter and Right Angle
- Cyclic Figures
Important Formulas and Theorems
Theorem 9
Corollary
Angles standing on the same chord are equal.
Diameter Theorem
The angle subtended by a diameter is 90°.
Exercise 5.6 Solutions
Question 1
In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of chord AB?
Solution
Question 2
Let A and B be two points on a circle with centre O.
(i) Are there points X, Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB?
Solution
(ii) Is it true that if ∠AXB = ∠AYB, then X and Y lie on the same side of the circle?
Solution
Equal angles may occur on opposite sides of chord AB.
Therefore,
[
\boxed{\text{X and Y need not lie on the same side of AB.}}
]
(iii) If ∠AXB = ∠AYB, and X and Y do not lie on the circle, does the circle through A, B and X also pass through Y?
Solution
Question 3
Find x in Fig. 5.26.
Common Mistakes
❌ Forgetting that the angle at the centre is twice the angle at the circumference.
❌ Assuming all equal angles must lie on the same side of the chord.
❌ Using the radius instead of the chord in Question 1.
❌ Taking x = 100° instead of 50° in Question 3.
Exam Tips
✔ Angles in the same segment are equal.
✔ Diameter always subtends a right angle.
✔ Draw neat diagrams before solving.
Practice MCQs
1. If an angle at the centre is 80°, then the angle at the circumference on the same chord is:
A. 20°
B. 40°
C. 80°
D. 160°
Answer: B
2. A diameter subtends an angle of:
A. 180°
B. 60°
C. 90°
D. 45°
Answer: C
3. If two angles stand on the same chord, then they are:
A. Complementary
B. Supplementary
C. Equal
D. Unequal
Answer: C
4. A chord subtends an angle of 35° at the circumference. The angle at the centre is:
A. 35°
B. 70°
C. 105°
D. 140°
Answer: B
Frequently Asked Questions (FAQs)
Q1. What is the relationship between the angle at the centre and the angle at the circumference?
The angle at the centre is twice the angle at the circumference standing on the same chord.
Q2. Why are angles in the same segment equal?
Because they subtend the same arc of the circle.
Q3. Which angle does a diameter subtend?
A right angle (90°).
Q4. What are concyclic points?
Points lying on the same circle are called concyclic points.
Key Takeaways
✅ Angle at centre = 2 × angle at circumference.
✅ Angles standing on the same chord are equal.
✅ Diameter subtends a right angle.
✅ Equal angles often help in proving points are concyclic.
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