Two chords AB and CD of a circle are each at distances 4 cm from the centre. Then,

Options

A. AB=CD.
B. AB is not equal to CD.
C. AB there is no relation between CD.
D. AB is greater than CD.

Explanation

To solve the problem, we need to show that the lengths of the two chords AB and CD are equal, given that both are at a distance of 4 cm from the center of the circle.

### Step-by-Step Solution:

1. **Draw the Circle and Chords**:
- Draw a circle with center O.
- Draw two chords AB and CD such that both are at a distance of 4 cm from the center O.

2. **Identify the Perpendiculars**:
- Draw a perpendicular line from the center O to chord AB, and let the foot of the perpendicular be point L.
- Similarly, draw a perpendicular line from O to chord CD, and let the foot of the perpendicular be point M.

3. **Label the Distances**:
- According to the problem, the distance OL (from O to chord AB) is 4 cm.
- The distance OM (from O to chord CD) is also 4 cm.

4. **Use the Properties of Triangles**:
- In triangle OLB, OB is the radius of the circle, and in triangle OMD, OD is also the radius of the circle. Therefore, OB = OD.
- Since OL = OM = 4 cm, we have OL = OM.

5. **Angles in the Triangles**:
- The angles ∠OLB and ∠OMD are both right angles (90 degrees) because we drew perpendiculars from the center to the chords.

6. **Congruent Triangles**:
- By the Side-Angle-Side (SAS) congruence criterion, triangles OLB and OMD are congruent since:
- OB = OD (radii of the circle),
- OL = OM (both are 4 cm),
- ∠OLB = ∠OMD = 90°.
- Therefore, we can conclude that triangle OLB ≅ triangle OMD.

7. **Corresponding Parts of Congruent Triangles**:
- Since the triangles are congruent, the corresponding sides are equal. Thus, LB = MD.

8. **Relate Chord Lengths to Segments**:
- From the property of circles, we know that the perpendicular from the center of a circle bisects the chord. Therefore:
- The length of chord AB can be expressed as AB = 2 * LB.
- The length of chord CD can be expressed as CD = 2 * MD.

9. **Equate the Lengths**:
- Since LB = MD, we can substitute:
- AB = 2 * LB and CD = 2 * MD ⇒ AB = CD.

10. **Conclusion**:
- Hence, we have proved that the lengths of the chords AB and CD are equal.

### Final Answer:
AB = CD