`PQ and RS` are two parallel chords of a circle. If `P Q = 30` cm, `RS = 16` cm and distance between `PQ and RS` is 23 cm, then find the radius of the circle.

Options

A. A)15
B. B)17
C. C)20
D. D)22

Answer

To find the radius of the circle given the two parallel chords PQ and RS, we can follow these steps:

### Step 1: Draw the Circle and Chords
Draw a circle with center O. Mark the two parallel chords PQ and RS such that PQ = 30 cm and RS = 16 cm.

**Hint:** Visualizing the problem with a diagram can help you understand the relationships between the components.

### Step 2: Identify Midpoints
Let M be the midpoint of chord PQ and N be the midpoint of chord RS. Since PQ = 30 cm, PM = MQ = 15 cm. For RS = 16 cm, RN = NS = 8 cm.

**Hint:** The midpoints of the chords will help in applying the Pythagorean theorem later.

### Step 3: Establish the Distance Between Chords
The distance between the two chords PQ and RS is given as 23 cm. Let the distance from the center O to chord PQ be x cm. Therefore, the distance from O to chord RS will be (23 - x) cm.

**Hint:** Setting up the distances correctly is crucial for applying the Pythagorean theorem.

### Step 4: Apply the Pythagorean Theorem
Using the right triangles formed by the radius and the perpendicular distances to the chords, we can set up two equations based on the Pythagorean theorem:
1. For triangle OMQ (with OM = x and MQ = 15):
\[ R^2 = x^2 + 15^2 \]
\[ R^2 = x^2 + 225 \]

2. For triangle ONR (with ON = 23 - x and NR = 8):
\[ R^2 = (23 - x)^2 + 8^2 \]
\[ R^2 = (23 - x)^2 + 64 \]

**Hint:** Remember that both expressions equal R², so you can set them equal to each other.

### Step 5: Set the Equations Equal
Now, set the two equations for R² equal to each other:
\[ x^2 + 225 = (23 - x)^2 + 64 \]

**Hint:** Expanding the squared term will help simplify the equation.

### Step 6: Expand and Simplify
Expanding the right side:
\[ x^2 + 225 = 529 - 46x + x^2 + 64 \]
Combine like terms:
\[ x^2 + 225 = 593 - 46x + x^2 \]
Subtract \(x^2\) from both sides:
\[ 225 = 593 - 46x \]
Rearranging gives:
\[ 46x = 593 - 225 \]
\[ 46x = 368 \]
\[ x = \frac{368}{46} = 8 \]

**Hint:** Solve for x carefully and ensure you perform arithmetic correctly.

### Step 7: Calculate the Radius
Now that we have x = 8, substitute it back into one of the equations for R²:
\[ R^2 = x^2 + 225 = 8^2 + 225 = 64 + 225 = 289 \]
Thus, the radius R is:
\[ R = \sqrt{289} = 17 \text{ cm} \]

**Hint:** Always take the positive square root when calculating the radius.

### Final Answer
The radius of the circle is **17 cm**.