In a circle of radius 17 cm, two parallel chord of length 30 cm and 16 cm are drawn, find the distance between the chords, if both the chords are :
On the opposite sides of the centre,

Answer

To solve the problem of finding the distance between two parallel chords in a circle, we can follow these steps:

### Step 1: Understand the Problem
We have a circle with a radius of 17 cm, and two parallel chords of lengths 30 cm and 16 cm. We need to find the distance between these two chords, which are on opposite sides of the center of the circle.

### Step 2: Draw the Diagram
Draw a circle with center O and radius 17 cm. Mark the two parallel chords AB and CD, where AB is 30 cm long and CD is 16 cm long. Label the midpoints of the chords as E and F, respectively.

### Step 3: Find Half-Lengths of the Chords
Since the chords are bisected by the perpendicular from the center:
- For chord AB:
\[ AE = \frac{30}{2} = 15 \text{ cm} \]
- For chord CD:
\[ CF = \frac{16}{2} = 8 \text{ cm} \]

### Step 4: Apply the Pythagorean Theorem
In triangle OAE (where O is the center, A is one endpoint of chord AB, and E is the midpoint of AB):
- The hypotenuse OA is the radius (17 cm).
- The side AE is 15 cm.
- We need to find OE (the perpendicular distance from O to chord AB).

Using the Pythagorean theorem:
\[ OA^2 = OE^2 + AE^2 \]
Substituting the known values:
\[ 17^2 = OE^2 + 15^2 \]
\[ 289 = OE^2 + 225 \]
\[ OE^2 = 289 - 225 = 64 \]
\[ OE = \sqrt{64} = 8 \text{ cm} \]

### Step 5: Repeat for Chord CD
Now, apply the Pythagorean theorem in triangle OCF (where O is the center, C is one endpoint of chord CD, and F is the midpoint of CD):
- The hypotenuse OC is also the radius (17 cm).
- The side CF is 8 cm.
- We need to find OF (the perpendicular distance from O to chord CD).

Using the Pythagorean theorem:
\[ OC^2 = OF^2 + CF^2 \]
Substituting the known values:
\[ 17^2 = OF^2 + 8^2 \]
\[ 289 = OF^2 + 64 \]
\[ OF^2 = 289 - 64 = 225 \]
\[ OF = \sqrt{225} = 15 \text{ cm} \]

### Step 6: Calculate the Distance Between the Chords
Since both chords are on opposite sides of the center, the total distance between the chords is:
\[ EF = OE + OF = 8 + 15 = 23 \text{ cm} \]

### Final Answer
The distance between the two chords is **23 cm**.

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