A 3cm long perpendicular is drawn from the centre of a circle to a 8cm long chord. Find the diameter of the circle.

Options

A. a. 6cm
B. b. 5cm
C. c. 4cm
D. d. 10cm

Answer

To solve the problem of finding the diameter of a circle given a perpendicular drawn from the center to a chord, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a circle with a chord of length 8 cm. A perpendicular line from the center of the circle to the chord is 3 cm long. We need to find the diameter of the circle.

2. **Draw the Diagram**:
- Let the center of the circle be point O.
- Let the chord be AB, with a length of 8 cm.
- The perpendicular from O to the chord AB meets the chord at point C.
- Since OC is perpendicular to AB, it divides the chord into two equal parts. Therefore, AC = CB = 4 cm (since 8 cm / 2 = 4 cm).

3. **Identify the Right Triangle**:
- We can now form a right triangle OAC, where:
- OC = 3 cm (the length of the perpendicular),
- AC = 4 cm (half the length of the chord),
- OA = r (the radius of the circle).

4. **Apply the Pythagorean Theorem**:
- According to the Pythagorean theorem, in triangle OAC:
\[ OA^2 = OC^2 + AC^2 \]
- Plugging in the values:
\[ OA^2 = 3^2 + 4^2 \]
\[ OA^2 = 9 + 16 \]
\[ OA^2 = 25 \]
- Taking the square root of both sides:
\[ OA = \sqrt{25} = 5 \text{ cm} \]

5. **Calculate the Diameter**:
- The diameter (D) of the circle is twice the radius:
\[ D = 2 \times OA = 2 \times 5 = 10 \text{ cm} \]

6. **Final Answer**:
- The diameter of the circle is 10 cm.