Circle Theorems: Central & Inscribed Angles
Theorem: Central angle is double the inscribed angle standing on the same arc. Prove that.
OR
Inscribed angle is half of the central angle standing on the same arc.
1. Given:
- A circle with center O.
- Central angle ∠AOB and inscribed angle ∠ACB standing on the same arc AB.
2. To Prove:
- ∠AOB = 2 ∠ACB
- OR
- ∠ACB = ½ ∠AOB
Figure 1: Standing on Same Arc AB
3. Proof:
| Statements | Reasons |
|---|---|
| 1. ∠AOB = AB | 1. Central angle and its opposite arc. |
| 2. ∠ACB = ½ AB | 2. Inscribed angle and its opposite arc. |
| 3. ∠ACB = ½ ∠AOB | 3. From statements 1 and 2. |
| 4. ∠AOB = 2 ∠ACB | 4. From statement 3. |
Proved.
Theorem: Inscribed angle standing on an arc is half of the central angle standing on an equal arc.
1. Given:
Arc AB = Arc CD in a circle with center O. Inscribed angle ∠APB stands on Arc AB and central angle ∠COD stands on Arc CD.
2. To Prove:
∠APB = ½ ∠COD (or ∠COD = 2 ∠APB)
Equal Arcs (Arc AB = Arc CD)
3. Proof:
| Statements | Reasons |
|---|---|
| 1. AB = CD | 1. Given equal arcs. |
| 2. ∠COD = CD | 2. Central angle and its opposite arc. |
| 3. ∠APB = ½ AB | 3. Inscribed angle and its opposite arc. |
| 4. ∠APB = ½ CD | 4. From statements 1 and 3. |
| 5. ∠APB = ½ ∠COD (or ∠COD = 2 ∠APB) | 5. From statements 2 and 4. |
Proved.
Experimental Verification
📌 Note (Special Examination Instruction):
- When proving double relation, write: ∠AOB = 2 ∠ACB
- When proving half relation, write: ∠ACB = ½ ∠AOB
1. Figures
Figure 1 (r = 3 cm)
Figure 2 (r = 3.5 cm)
2. To be Experimented
∠AOB = 2 ∠ACB OR ∠ACB = ½ ∠AOB
3. Table
| Fig No. | Central Angle (∠AOB) | Inscribed Angle (∠ACB) | Remarks / Relation |
|---|---|---|---|
| Figure 1 | 100° | 50° | ∠AOB = 2 ∠ACB (or ∠ACB = ½ ∠AOB) |
| Figure 2 | 120° | 60° | ∠AOB = 2 ∠ACB (or ∠ACB = ½ ∠AOB) |
4. Conclusion
Hence, it is experimentally verified that the central angle is double the inscribed angle standing on the same arc (or the inscribed angle is half of the central angle).
Dynamic Demonstration
Drag Point A, Point B, Point C, or Point O to dynamically alter arc sizes, circle position, and angles in real time.
Central Angle (∠AOB) = 0° |
Inscribed Angle (∠ACB) = 0°