A chord of length 24 cm is drawn in a circular of diameter 40 cm. Another chord of length 32 cm is drawn in the same circle parallel to 24 cm long chord. What is the minimum distance (in cm) between them ?

Options

A. 4
B. 2
C. 8
D. 3

Answer

To find the minimum distance between the two parallel chords in a circle, we can follow these steps:

### Step 1: Determine the radius of the circle
The diameter of the circle is given as 40 cm. Therefore, the radius (r) can be calculated as:
\[ r = \frac{\text{Diameter}}{2} = \frac{40 \text{ cm}}{2} = 20 \text{ cm} \]

### Step 2: Find the perpendicular distance from the center to each chord
For any chord in a circle, the perpendicular distance from the center of the circle to the chord can be determined using the formula:
\[ d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2} \]
where \( l \) is the length of the chord.

#### For the first chord (24 cm):
\[ d_1 = \sqrt{20^2 - \left(\frac{24}{2}\right)^2} = \sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16 \text{ cm} \]

#### For the second chord (32 cm):
\[ d_2 = \sqrt{20^2 - \left(\frac{32}{2}\right)^2} = \sqrt{20^2 - 16^2} = \sqrt{400 - 256} = \sqrt{144} = 12 \text{ cm} \]

### Step 3: Calculate the minimum distance between the two chords
Since both chords are parallel and on the same side of the center of the circle, the minimum distance between them can be calculated by finding the difference between their distances from the center:
\[ \text{Minimum Distance} = d_1 - d_2 = 16 \text{ cm} - 12 \text{ cm} = 4 \text{ cm} \]

### Final Answer:
The minimum distance between the two chords is **4 cm**.

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