If a circle with radius of 10 cm has two parallel chords 16 cm and 12 cm and they are one the same side of the centre of the circle, then the distance between the two parallel chords is

Options

A. 2 cm
B. 3 cm
C. 5 cm
D. 8 cm

Answer

To solve the problem of finding the distance between two parallel chords of lengths 16 cm and 12 cm in a circle with a radius of 10 cm, we will follow these steps:

### Step 1: Understand the Geometry
We have a circle with a radius of 10 cm and two parallel chords, AB (16 cm) and CD (12 cm), on the same side of the center O of the circle. We need to find the distance between these two chords.

### Step 2: Draw the Diagram
Draw a circle with center O. Mark the chords AB and CD parallel to each other. Label the points where the perpendiculars from O meet the chords as E and F, respectively.

### Step 3: Find the Lengths of the Halves of the Chords
Since AB is 16 cm long, each half (AE and BE) will be:
\[ AE = BE = \frac{16}{2} = 8 \text{ cm} \]

For chord CD, which is 12 cm long, each half (CF and DF) will be:
\[ CF = DF = \frac{12}{2} = 6 \text{ cm} \]

### Step 4: Use the Pythagorean Theorem
For triangle OAE:
- OA is the radius (10 cm).
- AE is half the length of chord AB (8 cm).
- We need to find OE (the perpendicular distance from O to chord AB).

Using the Pythagorean theorem:
\[ OA^2 = OE^2 + AE^2 \]
\[ 10^2 = OE^2 + 8^2 \]
\[ 100 = OE^2 + 64 \]
\[ OE^2 = 100 - 64 = 36 \]
\[ OE = \sqrt{36} = 6 \text{ cm} \]

For triangle OCF:
- OC is also the radius (10 cm).
- CF is half the length of chord CD (6 cm).
- We need to find OF (the perpendicular distance from O to chord CD).

Using the Pythagorean theorem:
\[ OC^2 = OF^2 + CF^2 \]
\[ 10^2 = OF^2 + 6^2 \]
\[ 100 = OF^2 + 36 \]
\[ OF^2 = 100 - 36 = 64 \]
\[ OF = \sqrt{64} = 8 \text{ cm} \]

### Step 5: Calculate the Distance Between the Chords
The distance between the two chords is the difference between OE and OF:
\[ \text{Distance} = OF - OE \]
\[ \text{Distance} = 8 \text{ cm} - 6 \text{ cm} = 2 \text{ cm} \]

### Final Answer
The distance between the two parallel chords is **2 cm**.

---