AB and CD are 2 chords of circle such that AB = 8 cm, CD = 10 cm distance b/w chord is 2 cm . If chords are on same side find radius

Answer

To find the radius of the circle given the lengths of two chords and the distance between them, we can follow these steps:

### Step 1: Understand the problem
We have two chords AB and CD in a circle, where:
- Length of chord AB = 8 cm
- Length of chord CD = 10 cm
- Distance between the two chords = 2 cm

We need to find the radius (R) of the circle.

### Step 2: Divide the chords into equal segments
Since a perpendicular drawn from the center of the circle to a chord bisects the chord, we can find the lengths of the segments:
- For chord AB:
- \( AL = LB = \frac{AB}{2} = \frac{8}{2} = 4 \) cm
- For chord CD:
- \( CK = KD = \frac{CD}{2} = \frac{10}{2} = 5 \) cm

### Step 3: Set up the coordinate system
Let:
- O be the center of the circle.
- K be the foot of the perpendicular from O to chord CD.
- L be the foot of the perpendicular from O to chord AB.
- The distance from O to chord AB is \( x + 2 \) cm (where \( x \) is the distance from O to chord CD).
- The distance from O to chord CD is \( x \) cm.

### Step 4: Apply the Pythagorean theorem
Using the Pythagorean theorem in triangles OAL and OCK:
1. For triangle OAL:
\[ OA^2 = AL^2 + OL^2 \implies R^2 = 4^2 + (x + 2)^2 \]
\[ R^2 = 16 + (x + 2)^2 \]

2. For triangle OCK:
\[ OC^2 = CK^2 + OK^2 \implies R^2 = 5^2 + x^2 \]
\[ R^2 = 25 + x^2 \]

### Step 5: Set the equations equal to each other
Since both expressions equal \( R^2 \):
\[ 16 + (x + 2)^2 = 25 + x^2 \]

### Step 6: Expand and simplify
Expanding the left side:
\[ 16 + (x^2 + 4x + 4) = 25 + x^2 \]
\[ x^2 + 4x + 20 = 25 + x^2 \]

Now, subtract \( x^2 \) from both sides:
\[ 4x + 20 = 25 \]

### Step 7: Solve for x
Rearranging gives:
\[ 4x = 25 - 20 \implies 4x = 5 \implies x = \frac{5}{4} \text{ cm} \]

### Step 8: Substitute x back to find R
Using the value of \( x \) in either equation for \( R^2 \):
Using \( R^2 = 25 + x^2 \):
\[ R^2 = 25 + \left(\frac{5}{4}\right)^2 = 25 + \frac{25}{16} \]
Convert 25 to a fraction:
\[ R^2 = \frac{400}{16} + \frac{25}{16} = \frac{425}{16} \]

### Step 9: Find R
Taking the square root:
\[ R = \sqrt{\frac{425}{16}} = \frac{\sqrt{425}}{4} = \frac{5\sqrt{17}}{4} \text{ cm} \]

### Final Answer
The radius of the circle is \( \frac{5\sqrt{17}}{4} \) cm.

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