Step 1: Understanding the Concept:
This is a classic geometry problem involving the relationship between the radius, a chord, and the perpendicular distance from the center to the chord. The key is to form a right-angled triangle.
Step 2: Key Formula or Approach:
1. A line drawn from the center of a circle perpendicular to a chord bisects the chord.
2. The Pythagorean theorem in a right-angled triangle: $a^2 + b^2 = c^2$, where c is the hypotenuse.
Step 3: Detailed Explanation:
Let the circle have center O and radius r. Let AB be the chord with length 6 cm.
Let M be the midpoint of AB. The distance from the center O to the chord AB is the length of the perpendicular segment OM, which is given as 4 cm.
Since the perpendicular from the center bisects the chord, the length of AM is half the length of AB.
\[ AM = \frac{AB}{2} = \frac{6}{2} = 3 \text{ cm} \]
Now, consider the triangle $\triangle OMA$. It is a right-angled triangle with the right angle at M.
The sides are:
OM = 4 cm (distance from center to chord)
AM = 3 cm (half the chord length)
OA = r (the radius of the circle, which is the hypotenuse)
Using the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]
\[ r^2 = 4^2 + 3^2 \]
\[ r^2 = 16 + 9 = 25 \]
\[ r = \sqrt{25} = 5 \text{ cm} \]
The question asks for the length of the diameter. The diameter (d) is twice the radius.
\[ d = 2r = 2 \times 5 = 10 \text{ cm} \]
Step 4: Final Answer:
The length of the diameter of the circle is 10 cm. Therefore, option (B) is correct.