AB and CD are two chords of a circle such that AB = 10 cm, CD - 24 cm and AB||CD. If the distance between AB and CD is 17 cm. Then, the radius of the circle is equal to

Options

A. 13
B. 169
C. 26
D. None of these

Answer

To find the radius of the circle given the conditions of the chords AB and CD, we can follow these steps:

### Step 1: Understand the given information
We have two parallel chords AB and CD in a circle:
- Length of chord AB = 10 cm
- Length of chord CD = 24 cm
- Distance between the two chords = 17 cm

### Step 2: Set up the diagram
Draw a circle with center O. Mark the chords AB and CD such that they are parallel. Let M be the midpoint of chord AB and N be the midpoint of chord CD.

### Step 3: Determine the distances
Since AB is 10 cm long, the distance from the center O to the chord AB (OM) can be expressed as:
- AM = MB = AB/2 = 10/2 = 5 cm

Similarly, since CD is 24 cm long, the distance from the center O to the chord CD (ON) can be expressed as:
- CN = ND = CD/2 = 24/2 = 12 cm

### Step 4: Define the distance between the chords
Let the distance from the center O to chord AB be x cm. Therefore, the distance from the center O to chord CD will be (17 - x) cm, since the total distance between the two chords is 17 cm.

### Step 5: Apply the Pythagorean theorem
Using the Pythagorean theorem for the radius (R) of the circle:
1. For chord AB:
\[ R^2 = OM^2 + AM^2 = x^2 + 5^2 = x^2 + 25 \quad \text{(Equation 1)} \]

2. For chord CD:
\[ R^2 = ON^2 + CN^2 = (17 - x)^2 + 12^2 = (17 - x)^2 + 144 \quad \text{(Equation 2)} \]

### Step 6: Expand Equation 2
Expanding Equation 2:
\[ R^2 = (17 - x)^2 + 144 = (289 - 34x + x^2) + 144 = x^2 - 34x + 433 \]

### Step 7: Set the equations equal
Since both expressions equal R², we can set them equal to each other:
\[ x^2 + 25 = x^2 - 34x + 433 \]

### Step 8: Simplify the equation
Cancelling \(x^2\) from both sides gives:
\[ 25 = -34x + 433 \]
Rearranging gives:
\[ 34x = 433 - 25 \]
\[ 34x = 408 \]
\[ x = \frac{408}{34} = 12 \]

### Step 9: Substitute x back into Equation 1
Now substitute \(x = 12\) back into Equation 1 to find R:
\[ R^2 = 12^2 + 25 = 144 + 25 = 169 \]
\[ R = \sqrt{169} = 13 \text{ cm} \]

### Final Answer
The radius of the circle is **13 cm**.

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