AB = 16 cm and CD = 12 cm are two parallel chords lie on same side of centre of a given circle. If distance between them is 2 cm then what is the radius of the circle?

Options

A. a)5 cm
B. b)10 cm
C. c)3 cm
D. d)12 cm

Answer

To find the radius of the circle given the lengths of two parallel chords and the distance between them, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of chord AB = 16 cm
- Length of chord CD = 12 cm
- Distance between the chords = 2 cm

2. **Draw the Circle and Chords:**
- Draw a circle with center O.
- Draw the two parallel chords AB and CD on the same side of the center O.

3. **Determine the Midpoints of the Chords:**
- Let P be the midpoint of chord AB. Since AB = 16 cm, then AP = PB = 8 cm.
- Let Q be the midpoint of chord CD. Since CD = 12 cm, then CQ = QD = 6 cm.

4. **Set Up the Right Triangles:**
- Draw perpendicular lines from the center O to the midpoints P and Q.
- The distance OP = x (the distance from the center to chord AB).
- The distance OQ = x + 2 (since the distance between the chords is 2 cm).

5. **Apply the Pythagorean Theorem:**
- For triangle OAP (right triangle):
\[ OA^2 = OP^2 + AP^2 \implies r^2 = x^2 + 8^2 \implies r^2 = x^2 + 64 \]
- For triangle OCQ (right triangle):
\[ OC^2 = OQ^2 + CQ^2 \implies r^2 = OQ^2 + 6^2 \implies r^2 = (x + 2)^2 + 36 \]

6. **Set the Equations Equal:**
- Since both expressions equal \( r^2 \):
\[ x^2 + 64 = (x + 2)^2 + 36 \]

7. **Expand and Simplify:**
- Expand the right side:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
- Thus, the equation becomes:
\[ x^2 + 64 = x^2 + 4x + 4 + 36 \]
- Simplifying gives:
\[ x^2 + 64 = x^2 + 4x + 40 \]
- Cancel \( x^2 \) from both sides:
\[ 64 = 4x + 40 \]

8. **Solve for x:**
- Rearranging gives:
\[ 64 - 40 = 4x \implies 24 = 4x \implies x = 6 \]

9. **Find the Radius:**
- Substitute \( x = 6 \) back into the equation for \( r^2 \):
\[ r^2 = x^2 + 64 = 6^2 + 64 = 36 + 64 = 100 \]
- Therefore, the radius \( r = \sqrt{100} = 10 \) cm.

### Final Answer:
The radius of the circle is **10 cm**.

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