The radius of a circle is 6 cm. The perpendicular distance from the centre of the circle to the chord which is 8 cm in length is

Options

A. `sqrt(5)` cm
B. `2 sqrt(5)` cm
C. `2 sqrt(7)` cm
D. `sqrt(7)` cm

Answer

To solve the problem, we will follow these steps:

1. **Understand the given information**: We have a circle with a radius of 6 cm, and there is a chord of length 8 cm. We need to find the perpendicular distance from the center of the circle to the chord.

2. **Draw the diagram**:
- Draw a circle with center O.
- Draw a chord AB of length 8 cm.
- Draw a perpendicular line from the center O to the chord AB, meeting the chord at point M.

3. **Identify the segments**:
- Since OM is perpendicular to AB, it bisects the chord. Therefore, AM = MB = 8 cm / 2 = 4 cm.

4. **Apply the Pythagorean theorem**:
- In triangle OMA, we can apply the Pythagorean theorem.
- Here, OA is the radius (6 cm), AM is half the chord length (4 cm), and OM is the perpendicular distance we want to find (let's denote it as h).

According to the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]

Substituting the known values:
\[ 6^2 = h^2 + 4^2 \]

5. **Calculate the squares**:
- \( 6^2 = 36 \)
- \( 4^2 = 16 \)

So, we have:
\[ 36 = h^2 + 16 \]

6. **Rearrange the equation**:
\[ h^2 = 36 - 16 \]
\[ h^2 = 20 \]

7. **Find h**:
\[ h = \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \]

8. **Conclusion**: The perpendicular distance from the center of the circle to the chord is \( 2\sqrt{5} \) cm.