Two parallel chords on the same sideof the centre of a circle are 12 cm and 20 cm long the radius of the circle is `5sqrt(13)`cm. What is the distance (in cm) between the chord?

Options

A. `2`
B. `3`
C. `2.5`
D. `1.5`

Answer

To solve the problem, we need to find the distance between two parallel chords in a circle. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Problem
We have a circle with a radius of \( 5\sqrt{13} \) cm, and two parallel chords of lengths 12 cm and 20 cm. We need to find the distance between these two chords.

### Step 2: Drawing the Circle and Chords
1. Draw a circle with center \( O \).
2. Draw two parallel chords \( AB \) and \( CD \) such that \( AB = 20 \) cm and \( CD = 12 \) cm. Both chords are on the same side of the center \( O \).

### Step 3: Finding the Distances from the Center to Each Chord
Using the property that the perpendicular from the center of the circle to a chord bisects the chord, we can find the distances from the center \( O \) to each chord.

#### For Chord \( AB \):
1. Since \( AB = 20 \) cm, the half-length \( AM = MB = 10 \) cm.
2. In triangle \( OAM \) (where \( M \) is the midpoint of chord \( AB \)):
- Hypotenuse \( OA = 5\sqrt{13} \) cm (radius).
- Base \( AM = 10 \) cm.
- Let \( OC \) be the perpendicular distance from \( O \) to chord \( AB \).

Using the Pythagorean theorem:
\[ OA^2 = OC^2 + AM^2 \]
\[ (5\sqrt{13})^2 = OC^2 + 10^2 \]
\[ 325 = OC^2 + 100 \]
\[ OC^2 = 325 - 100 = 225 \]
\[ OC = \sqrt{225} = 15 \text{ cm} \]

#### For Chord \( CD \):
1. Since \( CD = 12 \) cm, the half-length \( CN = ND = 6 \) cm.
2. In triangle \( OCN \) (where \( N \) is the midpoint of chord \( CD \)):
- Hypotenuse \( OC = 5\sqrt{13} \) cm (radius).
- Base \( CN = 6 \) cm.
- Let \( OD \) be the perpendicular distance from \( O \) to chord \( CD \).

Using the Pythagorean theorem:
\[ OC^2 = OD^2 + CN^2 \]
\[ (5\sqrt{13})^2 = OD^2 + 6^2 \]
\[ 325 = OD^2 + 36 \]
\[ OD^2 = 325 - 36 = 289 \]
\[ OD = \sqrt{289} = 17 \text{ cm} \]

### Step 4: Finding the Distance Between the Chords
Now, we can find the distance between the two chords:
\[ \text{Distance between the chords} = OD - OC = 17 - 15 = 2 \text{ cm} \]

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

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