If a chord of length 16 cm is at a distance of 15 cm from the centre of the circle, then the length of the chord of the same circle which is at a distance of 8 cm from the centre is equal to

Options

A. 10 cm
B. 20 cm
C. 30 cm
D. 40 cm

Answer

To solve the problem step by step, we need to find the length of a chord in a circle given certain conditions. Here’s how we can approach it:

### Step 1: Understand the Given Information
We have a circle with:
- A chord \( AB \) of length \( 16 \, \text{cm} \) that is \( 15 \, \text{cm} \) away from the center \( O \).
- We need to find the length of another chord \( CD \) that is \( 8 \, \text{cm} \) away from the center \( O \).

### Step 2: Draw the Diagram
Draw a circle and label the center as \( O \). Draw the chord \( AB \) such that the perpendicular distance from \( O \) to \( AB \) is \( 15 \, \text{cm} \).

### Step 3: Use the Properties of Chords
The perpendicular from the center of the circle to a chord bisects the chord. Therefore, if \( AB = 16 \, \text{cm} \), then:
- \( AC = CB = \frac{16}{2} = 8 \, \text{cm} \)

### Step 4: Apply the Pythagorean Theorem
In triangle \( OAC \):
- \( OA \) is the radius \( r \) of the circle.
- \( OC = 15 \, \text{cm} \) (the distance from the center to the chord).
- \( AC = 8 \, \text{cm} \) (half the length of the chord).

Using the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
Substituting the known values:
\[ r^2 = 15^2 + 8^2 \]
\[ r^2 = 225 + 64 = 289 \]
\[ r = \sqrt{289} = 17 \, \text{cm} \]

### Step 5: Find the Length of the New Chord
Now, we need to find the length of chord \( CD \) which is \( 8 \, \text{cm} \) away from the center \( O \).

In triangle \( OED \) (where \( E \) is the midpoint of chord \( CD \)):
- \( OE = 8 \, \text{cm} \) (the distance from the center to the chord).
- \( OD = r = 17 \, \text{cm} \).

Using the Pythagorean theorem again:
\[ OD^2 = OE^2 + ED^2 \]
Substituting the known values:
\[ 17^2 = 8^2 + ED^2 \]
\[ 289 = 64 + ED^2 \]
\[ ED^2 = 289 - 64 = 225 \]
\[ ED = \sqrt{225} = 15 \, \text{cm} \]

### Step 6: Calculate the Length of Chord \( CD \)
Since \( CD \) is bisected by \( OE \):
\[ CD = 2 \times ED = 2 \times 15 = 30 \, \text{cm} \]

### Final Answer
The length of the chord \( CD \) which is at a distance of \( 8 \, \text{cm} \) from the center is \( 30 \, \text{cm} \).

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