A chord of length 6 cm is 4 cm away from the centre of circle. Find the diameter of the circle.

Options

A. 10 cm
B. 6 cm
C. 5 cm
D. 8 cm

Answer

To find the diameter of the circle given that a chord of length 6 cm is 4 cm away from the center of the circle, we can follow these steps:

### Step 1: Understand the Geometry
We have a circle with center O. There is a chord AB that is 6 cm long, and the perpendicular distance from the center O to the chord AB is 4 cm. Let C be the point where the perpendicular from O meets the chord AB.

### Step 2: Bisect the Chord
Since the perpendicular from the center of the circle to the chord bisects the chord, we can say:
- AC = BC = half of AB
- Therefore, AC = BC = 6 cm / 2 = 3 cm.

### Step 3: Set Up the Right Triangle
Now, we can form a right triangle OAC where:
- OC = 4 cm (the distance from the center to the chord),
- AC = 3 cm (half the length of the chord),
- OA = radius of the circle (which we need to find).

### Step 4: Apply the Pythagorean Theorem
Using the Pythagorean theorem in triangle OAC:
\[ OA^2 = OC^2 + AC^2 \]
Substituting the known values:
\[ OA^2 = 4^2 + 3^2 \]
\[ OA^2 = 16 + 9 \]
\[ OA^2 = 25 \]
Taking the square root:
\[ OA = \sqrt{25} = 5 \text{ cm} \]

### Step 5: Find the Diameter
The diameter of the circle is twice the radius:
\[ \text{Diameter} = 2 \times OA = 2 \times 5 = 10 \text{ cm} \]

### Final Answer
The diameter of the circle is **10 cm**.

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