A chord of length 24 cm is at a distance of 5 cm from the centre of the circle. Find the length of the chord of the same circle which is at a distance of 12 cm from the centre.

Answer

To solve the problem step by step, we will use the properties of circles and the Pythagorean theorem.

### Step 1: Understand the Problem
We have a chord \( CD \) of length 24 cm that is at a distance of 5 cm from the center \( O \) of the circle. We need to find the length of another chord \( AB \) that is at a distance of 12 cm from the center \( O \).

### Step 2: Draw the Diagram
Draw a circle with center \( O \). Mark the chord \( CD \) such that the perpendicular distance from \( O \) to \( CD \) is 5 cm. Let \( Q \) be the foot of the perpendicular from \( O \) to \( CD \). Since \( CD \) is 24 cm long, \( CQ = 12 \) cm and \( DQ = 12 \) cm (because \( Q \) is the midpoint of \( CD \)).

### Step 3: Apply the Pythagorean Theorem
In triangle \( OCQ \):
- \( OC \) is the radius of the circle.
- \( OQ = 5 \) cm (the distance from the center to the chord).
- \( CQ = 12 \) cm (half the length of the chord).

Using the Pythagorean theorem:
\[ OC^2 = OQ^2 + CQ^2 \]
Substituting the known values:
\[ OC^2 = 5^2 + 12^2 \]
\[ OC^2 = 25 + 144 = 169 \]
\[ OC = \sqrt{169} = 13 \text{ cm} \]

### Step 4: Find the Length of the New Chord \( AB \)
Now, we need to find the length of the chord \( AB \) which is at a distance of 12 cm from the center \( O \). Let \( P \) be the foot of the perpendicular from \( O \) to \( AB \).

In triangle \( OAP \):
- \( OA = OC = 13 \) cm (the radius).
- \( OP = 12 \) cm (the distance from the center to the chord).

Using the Pythagorean theorem again:
\[ OA^2 = OP^2 + AP^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + AP^2 \]
\[ 169 = 144 + AP^2 \]
\[ AP^2 = 169 - 144 = 25 \]
\[ AP = \sqrt{25} = 5 \text{ cm} \]

### Step 5: Calculate the Full Length of Chord \( AB \)
Since \( P \) is the midpoint of \( AB \), we have:
\[ AB = AP + PB = AP + AP = 2 \times AP = 2 \times 5 = 10 \text{ cm} \]

### Final Answer
The length of the chord \( AB \) is \( 10 \) cm.

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