Three circles of diameter 24 cm each touch each other externally. Find the shortest distance from the centre of one circle to the line joining the centres of the other two circles. (in cm.)

Options

A. `12sqrt(3)`
B. `24sqrt(3)`
C. `24sqrt(6)`
D. `12sqrt(6)`

Answer

To solve the problem, we need to find the shortest distance from the center of one circle to the line joining the centers of the other two circles. Here’s a step-by-step solution:

### Step 1: Understand the configuration of the circles
We have three circles, each with a diameter of 24 cm. Therefore, the radius of each circle is:
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 2: Identify the centers of the circles
Let’s denote the centers of the three circles as \( A \), \( B \), and \( C \). Since the circles touch each other externally, the distance between the centers of any two circles is equal to the sum of their radii:
\[ AB = AC = BC = 12 + 12 = 24 \text{ cm} \]

### Step 3: Form an equilateral triangle
The centers \( A \), \( B \), and \( C \) form an equilateral triangle with each side measuring 24 cm.

### Step 4: Find the height of the equilateral triangle
To find the shortest distance from the center of one circle (say circle \( A \)) to the line joining the centers of the other two circles (line \( BC \)), we need to calculate the height of the equilateral triangle \( ABC \).

The formula for the height \( h \) of an equilateral triangle with side length \( a \) is given by:
\[ h = \frac{\sqrt{3}}{2} \times a \]
In our case, \( a = 24 \) cm, so:
\[ h = \frac{\sqrt{3}}{2} \times 24 = 12\sqrt{3} \text{ cm} \]

### Step 5: Conclusion
The shortest distance from the center of circle \( A \) to the line joining the centers of circles \( B \) and \( C \) is:
\[ \text{Shortest Distance} = 12\sqrt{3} \text{ cm} \]

### Final Answer
Thus, the shortest distance from the center of one circle to the line joining the centers of the other two circles is \( 12\sqrt{3} \) cm.

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