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 :

Options

A. 8 cm
B. 10 cm
C. 15 cm
D. 17 cm

Answer

To find the distance of the chord AB from the center of the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Diameter \( AD = 34 \, \text{cm} \)
- Chord \( AB = 30 \, \text{cm} \)

2. **Calculate the Radius of the Circle:**
\[ \text{Radius} (R) = \frac{\text{Diameter}}{2} = \frac{34}{2} = 17 \, \text{cm} \]

3. **Determine the Midpoint of the Chord:**
- Let \( O \) be the center of the circle.
- The midpoint \( P \) of the chord \( AB \) divides it into two equal segments:
\[ AP = PB = \frac{AB}{2} = \frac{30}{2} = 15 \, \text{cm} \]

4. **Apply the Pythagorean Theorem:**
- In the right triangle \( AOP \):
- \( AO \) is the radius \( R = 17 \, \text{cm} \)
- \( AP = 15 \, \text{cm} \)
- Let \( OP \) be the distance from the center \( O \) to the chord \( AB \) (which we need to find).
- According to the Pythagorean theorem:
\[ AO^2 = AP^2 + OP^2 \]
Substituting the known values:
\[ 17^2 = 15^2 + OP^2 \]

5. **Calculate the Squares:**
\[ 289 = 225 + OP^2 \]

6. **Isolate \( OP^2 \):**
\[ OP^2 = 289 - 225 = 64 \]

7. **Find \( OP \):**
\[ OP = \sqrt{64} = 8 \, \text{cm} \]

### Conclusion:
The distance of the chord \( AB \) from the center of the circle is \( 8 \, \text{cm} \).