1. What are Congruent Circles?
Congruent circles are circles that have the same radius. They may have different centres or be located in different positions, but if their radii are equal, the circles are congruent.
If two circles have radii \(r_1\) and \(r_2\), then they are congruent when:
\(r_1 = r_2\)
2. Key Features of Congruent Circles
Congruent circles share several geometric properties due to having equal radii.
2.1. Equal Radii
The only condition for congruent circles is that their radii must be equal. Their centres may be anywhere on the plane.
2.2. Equal Circumference and Area
If the radii of two circles are equal, then:
\(\text{Circumference}_1 = \text{Circumference}_2\)
\(\text{Area}_1 = \text{Area}_2\)
2.3. Same Shape and Size
Congruent circles are identical in size. One can be perfectly placed on top of the other by shifting without resizing or rotating.
3. Difference Between Concentric and Congruent Circles
Students often confuse these two concepts, but they describe different ideas.
3.1. Concentric Circles
Concentric circles share the same centre but have different radii.
3.2. Congruent Circles
Congruent circles have the same radius but may have different centres.
4. Congruent Arcs and Chords
In congruent circles, if the arcs subtend the same angle, the arcs and corresponding chords are congruent as well.
4.1. Equal Angles → Equal Arcs
If central angles are the same in congruent circles, the arcs they cut off are also equal.
4.2. Equal Arcs → Equal Chords
In congruent circles, equal arcs correspond to equal chords and vice versa.
5. Real-Life Examples
Congruent circles appear whenever identical circular shapes are used:
- Equal-sized coins
- Wheels of the same model of bike
- Buttons of identical size
- Circular lids produced in the same mould