AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, then the distance of AB from the centre of the circle is equal to :

Question

Accepted Answer

To determine the distance of chord AB from the center of the circle, we can use the properties of a circle and the Pythagorean theorem. Given that AD is the diameter of the circle and AB is a chord, we can solve the problem as follows:

1. **Understanding the Circle and Chord:**
   - AD is the diameter of the circle, which means the center of the circle, O, lies on AD. Therefore, AO = OD = 17 cm (since AD = 34 cm, the radius is half of the diameter).
   - AB is a chord of the circle, and we need to find the perpendicular distance from the center O to the chord AB.

2. **Using the Perpendicular from the Center:**
   - Let the perpendicular from O to AB intersect AB at point M. OM is the distance we need to find.
   - In a circle, the perpendicular from the center to a chord bisects the chord. Therefore, AM = MB = 15 cm (since AB = 30 cm).

3. **Applying the Pythagorean Theorem:**
   - In the right triangle OMA, we have:
     - OA = 17 cm (radius of the circle)
     - AM = 15 cm (half of the chord)
     - OM is the perpendicular distance we need to find.

Using the Pythagorean theorem:
   $$
   OA^2 = OM^2 + AM^2
   $$
   $$
   17^2 = OM^2 + 15^2
   $$
   $$
   289 = OM^2 + 225
   $$
   $$
   OM^2 = 289 - 225
   $$
   $$
   OM^2 = 64
   $$
   $$
   OM = \sqrt{64} = 8 \text{ cm}
   $$

4. **Conclusion:**
   - The distance of the chord AB from the center of the circle is 8 cm, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (15 cm):** This is incorrect because 15 cm is the length of AM, which is half of the chord AB, not the distance from the center to the chord.
- **Option 2 (17 cm):** This is incorrect because 17 cm is the radius of the circle, not the perpendicular distance from the center to the chord.
- **Option 3 (4 cm):** This is incorrect because the calculated perpendicular distance from the center to the chord is 8 cm, not 4 cm.

Thus, the correct answer is indeed Option 4, which is 8 cm.