In the given diagram 'O is the centre of the circle and AB is parallel to CD. AB = 24 cm and distance between he chord AB and CD is 17 cm. It the radius of the circle is 13 cm, find the length of the chord CD.

Answer

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

### Step 1: Understand the Given Information
- We have a circle with center O.
- Chord AB is parallel to chord CD.
- Length of chord AB (AB) = 24 cm.
- Distance between the chords AB and CD = 17 cm.
- Radius of the circle (OA) = 13 cm.

### Step 2: Find the Half-Length of Chord AB
Since AB is a chord, we can find the midpoint M of AB. The length of AM will be half of AB.
\[ AM = \frac{AB}{2} = \frac{24 \text{ cm}}{2} = 12 \text{ cm} \]

### Step 3: Apply the Pythagorean Theorem in Triangle OAM
In triangle OAM, we can apply the Pythagorean theorem:
\[ OA^2 = AM^2 + OM^2 \]
Where:
- OA = radius = 13 cm
- AM = 12 cm
- OM is the perpendicular distance from O to chord AB.

Substituting the known values:
\[ 13^2 = 12^2 + OM^2 \]
\[ 169 = 144 + OM^2 \]
\[ OM^2 = 169 - 144 = 25 \]
\[ OM = \sqrt{25} = 5 \text{ cm} \]

### Step 4: Find the Distance ON
We know the distance from chord AB to chord CD is 17 cm. Therefore, we can find ON:
\[ ON = \text{Distance between chords} - OM \]
\[ ON = 17 \text{ cm} - 5 \text{ cm} = 12 \text{ cm} \]

### Step 5: Apply the Pythagorean Theorem in Triangle OCN
Now, we will apply the Pythagorean theorem in triangle OCN:
\[ OC^2 = ON^2 + CN^2 \]
Where:
- OC = radius = 13 cm
- ON = 12 cm
- CN is the half-length of chord CD.

Substituting the known values:
\[ 13^2 = 12^2 + CN^2 \]
\[ 169 = 144 + CN^2 \]
\[ CN^2 = 169 - 144 = 25 \]
\[ CN = \sqrt{25} = 5 \text{ cm} \]

### Step 6: Find the Length of Chord CD
Since CN is half of chord CD, we can find the full length of chord CD:
\[ CD = 2 \times CN = 2 \times 5 \text{ cm} = 10 \text{ cm} \]

### Final Answer
The length of chord CD is **10 cm**.

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