Step 1: Understanding the Concept:
This problem explores the fundamental geometric relationship between a circle's center, its chords, and its radius.
The core theorem states that a perpendicular dropped from the center of a circle to any chord bisects that chord into two equal halves.
This geometric property allows us to visualize a right-angled triangle within the circle's structure.
In this specific scenario, the perpendicular distance serves as one leg of the triangle, and half of the chord length serves as the second leg.
The hypotenuse of this triangle is the line segment connecting the center to the circumference, which is the radius.
By determining the radius through algebraic calculation, we can eventually derive the diameter, which is the total width of the circle passing through the center.
Step 2: Key Formula or Approach:
The solution follows a logical progression using the Pythagorean Theorem and circle theorems.
1. Chord Bisector Theorem: If \( OM \perp AB \), then \( AM = MB \).
2. Pythagorean Theorem: For any right triangle, \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
3. In circle terms: \( (\text{Radius})^2 = (\text{Perpendicular Distance})^2 + (\text{Half-Chord})^2 \).
4. Diameter calculation: \( \text{Diameter} = 2 \times \text{Radius} \).
Step 2: Detailed Explanation:
Let us represent the circle with center \( O \).
Let the chord be labeled as \( AB \), with a given total length of 24 cm.
The perpendicular distance from center \( O \) to the chord \( AB \) is given as 5 cm. Let's call the point where this perpendicular meets the chord \( M \).
Thus, \( OM = 5 \) cm and \( \angle OMA = 90^\circ \).
According to the chord property, point \( M \) divides \( AB \) into two equal segments.
\[ AM = \frac{AB}{2} = \frac{24}{2} = 12 \text{ cm.} \]
We now focus on the right-angled triangle \( \triangle OMA \).
The distance \( OA \) represents the radius \( r \) of the circle because it connects the center to a point on the boundary.
Applying the Pythagorean theorem to \( \triangle OMA \):
\[ OA^2 = OM^2 + AM^2 \]
Substituting the numerical values:
\[ r^2 = 5^2 + 12^2 \]
\[ r^2 = 25 + 144 \]
\[ r^2 = 169 \]
To find the radius \( r \), we compute the square root of 169:
\[ r = \sqrt{169} = 13 \text{ cm.} \]
The problem asks for the diameter of the circle, not just the radius.
The diameter \( d \) is always twice the length of the radius:
\[ d = 2 \times r = 2 \times 13 = 26 \text{ cm.} \]
The logic holds because the radius is the distance from the center to the edge, while the diameter is the distance from edge to edge through the center.
Step 3: Final Answer:
The radius is 13 cm, and the diameter is 26 cm.
Therefore, the correct choice is Option (A).