Chords PQ and RS of a circle intersect at a point T at right angles to each other. If the segments PT, TQ and TS are of length 2, 6 and 3 units respectively then the diameter of the circle is

(a) 8

(b) 13

(c) sqrt(65)

(d) sqrt(47)

Given that chords \( PQ \) and \( RS \) of a circle intersect at point \( T \) at right angles to each other, and the segments \( PT = 2 \) units, \( TQ = 6 \) units, and \( TS = 3 \) units respectively, we need to determine the diameter of the circle from the given options.

### Steps to Solve:

1. **Use the Power of a Point Theorem:**
   The power of a point theorem states that if two chords intersect at a point, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord:
   \[
   PT \times TQ = RT \times TS
   \]
   Substituting the given lengths:
   \[
   2 \times 6 = RT \times 3
   \]
   \[
   12 = 3 \times RT
   \]
   \[
   RT = \frac{12}{3} = 4 \text{ units}
   \]

2. **Calculate the Lengths of the Chords:**
   - \( PQ = PT + TQ = 2 + 6 = 8 \) units.
   - \( RS = RT + TS = 4 + 3 = 7 \) units.

3. **Find the Length of the Diameter:**
   Since the chords \( PQ \) and \( RS \) intersect at right angles, we can use the fact that the sum of the squares of the distances from \( T \) to the ends of the chords equals the square of the diameter:
   \[
   \text{Diameter}^2 = PT^2 + TQ^2 + RT^2 + TS^2
   \]

   Substituting the values:
   \[
   \text{Diameter}^2 = 2^2 + 6^2 + 4^2 + 3^2
   \]
   \[
   \text{Diameter}^2 = 4 + 36 + 16 + 9
   \]
   \[
   \text{Diameter}^2 = 65
   \]
   \[
   \text{Diameter} = \sqrt{65} \text{ units}
   \]

### Answer:
The correct diameter of the circle is \( \boxed{\sqrt{65}} \) units, corresponding to option **(c)**.

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Do you have any further questions, or would you like more details on any part of this solution?

**Related Questions:**
1. What is the significance of right-angle intersection in a circle's chords?
2. How does the power of a point theorem apply in different geometric configurations?
3. What is the relationship between the diameter and the chord lengths in a circle?
4. How can we find the radius if only the length of one chord is given?
5. What would be the approach if the chords did not intersect at right angles?

**Tip:** When solving geometry problems, always check if special theorems (like the Power of a Point) can simplify your calculations.

Learn how to calculate the diameter of a circle when given intersecting chords at right angles. This geometry problem involves applying the Power of a Point Theorem and calculating chord lengths to find the diameter. Suitable for high school students.

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