In a circle of radius 17 cm a chord is at a distance of 15 cm form the centre of the circle. What is the length of the chord ?

Options

A. 15 cm
B. 12 cm
C. 8 cm
D. 16 cm

Answer

To find the length of the chord in a circle, we can use the following steps:

### Step 1: Understand the Problem
We have a circle with a radius \( r = 17 \) cm and a chord that is at a distance \( d = 15 \) cm from the center of the circle. We need to find the length of this chord.

### Step 2: Draw a Diagram
Draw a circle with center \( O \) and radius \( 17 \) cm. Mark the chord \( AB \) and the perpendicular distance from the center \( O \) to the chord \( AB \) as \( OM \), where \( M \) is the midpoint of the chord.

### Step 3: Apply the Pythagorean Theorem
In the right triangle \( OMA \):
- \( OA \) is the radius of the circle, which is \( 17 \) cm.
- \( OM \) is the distance from the center to the chord, which is \( 15 \) cm.
- \( AM \) is half the length of the chord, which we will denote as \( x \).

According to the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]

Substituting the known values:
\[ 17^2 = 15^2 + x^2 \]

### Step 4: Calculate the Values
Calculating the squares:
\[ 289 = 225 + x^2 \]

Now, isolate \( x^2 \):
\[ x^2 = 289 - 225 \]
\[ x^2 = 64 \]

Taking the square root of both sides:
\[ x = \sqrt{64} = 8 \text{ cm} \]

### Step 5: Find the Length of the Chord
Since \( x \) is half the length of the chord \( AB \):
\[ AB = 2 \times x = 2 \times 8 = 16 \text{ cm} \]

### Final Answer
The length of the chord is \( 16 \) cm.

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