Central Angles & Circle Theorems
Interactive Geometry Lab for Class 10 / SEE Preparation
Proof Hint: Draw equal arcs $AB$ and $CD$, join radii $OA, OB, OC, OD$ to form central angles $\angle AOB$ and $\angle COD$.
SEE Exam Solving Tip: Always remember that a central angle is directly equal to its intercepted arc ($\angle AOB \stackrel{\circ}{=} \widehat{AB}$).
SEE Exam Solving Tip: Always remember that a central angle is directly equal to its intercepted arc ($\angle AOB \stackrel{\circ}{=} \widehat{AB}$).
General Rules & Concept Relations
1. Central Angle & Arc Degree Relation
A Central Angle is an angle whose vertex is at the center of the circle ($O$) and whose sides are radii intersecting the circumference.
- Fundamental Relation: The measure of a central angle is directly equal to the degree measure of its intercepted arc.
- Mathematical Notation: $\angle AOB \stackrel{\circ}{=} \widehat{AB}$
- Equal Arcs Rule: If two arcs in a circle (or congruent circles) are equal in length ($\widehat{AB} = \widehat{CD}$), then their corresponding central angles are also equal ($\angle AOB = \angle COD$).
2. Inscribed Angle vs Central Angle Relation
An Inscribed Angle is formed when two chords meet at a point on the circumference.
- Inscribed Angle Rule: An inscribed angle is equal to half of the central angle standing on the same arc.
- Formula: $\angle APB \stackrel{\circ}{=} \frac{1}{2} \angle AOB \stackrel{\circ}{=} \frac{1}{2} \widehat{AB}$
Theorem Statement
Prove that central angles standing on equal arcs of a circle are equal.
Given:
$O$ is the center of the circle.
Arc $\widehat{AB} = \text{Arc } \widehat{CD}$. Central angles $\angle AOB$ and $\angle COD$ stand on arc $\widehat{AB}$ and arc $\widehat{CD}$ respectively.
To Prove: $\angle AOB = \angle COD$
Table
| Statements | Reasons |
|---|---|
| 1. $\angle AOB \stackrel{\circ}{=} \widehat{AB}$ | 1. A central angle is equal to its intercepted arc. |
| 2. $\angle COD \stackrel{\circ}{=} \widehat{CD}$ | 2. Same as reason 1. |
| 3. $\widehat{AB} = \widehat{CD}$ | 3. Given. |
| 4. $\angle AOB = \angle COD$ | 4. From statements 1, 2, and 3. |
Proved.
Theoretical Figure
Question: Verify experimentally that the central angles standing on equal arcs are equal.
Two circles of different measurements having radii greater than 3 cm are required.
1. Figures:
Figure 1 ($r = 3\text{ cm}$, $\widehat{AB} = \widehat{CD}$)
Figure 2 ($r = 4\text{ cm}$, $\widehat{AB} = \widehat{CD}$)
To be experimented: $\angle AOB = \angle COD$
Observation Table
| Figure No. | $\angle AOB$ | $\angle COD$ | Result |
|---|---|---|---|
| 1 | $60^\circ$ | $60^\circ$ | $\angle AOB = \angle COD$ |
| 2 | $75^\circ$ | $75^\circ$ | $\angle AOB = \angle COD$ |
Conclusion: Hence, it is experimentally verified that the central angles standing on equal arcs are equal.
Frequently Asked Questions (FAQ) & Practical Guide
Q1: Why must the radius of the circle be at least 3 cm (Radius > 3 cm) for experimental verification?
In geometry practical work, drawing a circle with a radius smaller than 3 cm creates a congested figure. When the radius is smaller than 3 cm, measuring angles using a protractor becomes extremely difficult and prone to human reading errors. Therefore, it is mandatory to use radii greater than 3 cm (e.g., $r = 3\text{ cm}$ for Figure 1 and $r = 4\text{ cm}$ for Figure 2) in examinations.
Q2: How to draw the circles step-by-step for Experimental Verification?
Follow these step-by-step instructions to draw accurate figures:
- Step 1: Take a ruler and set your compass width to $3\text{ cm}$. Mark a center point $O$ on paper and draw the first circle (Figure 1).
- Step 2: Set your compass width to $4\text{ cm}$ and draw the second circle (Figure 2) at another location.
- Step 3: Using a compass or thread, mark an arc $\widehat{AB}$ on the circle. Without changing the compass span, mark another equal arc $\widehat{CD}$ on the same circle so that $\widehat{AB} = \widehat{CD}$.
- Step 4: Join radii $OA, OB$ and $OC, OD$ using a ruler to form central angles $\angle AOB$ and $\angle COD$.
Q3: How to accurately measure the angles with a Protractor?
- Place the center hole/crosshair of your protractor exactly on the circle’s center point $O$.
- Align the baseline ($0^\circ$ line) of the protractor along the radius line $OA$.
- Read the degree value on the inner/outer scale where radius line $OB$ passes. This gives the value of $\angle AOB$.
- Repeat the process for $\angle COD$ by placing the protractor baseline along $OC$. Record both measured values in the observation table.