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The problem involves finding the length of the chord \( PQ \) in a circle where:

- \( O \) is the center of the circle.
- The radius of the circle is \( 18 \, \text{cm} \).
- The perpendicular distance from the center \( O \) to the chord \( PQ \) is \( 5 \, \text{cm} \).

We can solve this using the Pythagorean theorem since the perpendicular from the center of a circle to a chord bisects the chord.

### Step-by-step solution:
1. Let \( M \) be the midpoint of the chord \( PQ \). The perpendicular from \( O \) to \( PQ \) is \( OM = 5 \, \text{cm} \).
   
2. The radius \( OP = 18 \, \text{cm} \), and \( OM \) is perpendicular to \( PQ \). Therefore, the right triangle \( OMP \) has:
   - \( OM = 5 \, \text{cm} \) (one leg),
   - \( OP = 18 \, \text{cm} \) (hypotenuse),
   - \( MP \) (half of the chord) is the unknown we need to find.

3. Applying the Pythagorean theorem:
   \[
   OP^2 = OM^2 + MP^2
   \]
   Substituting the known values:
   \[
   18^2 = 5^2 + MP^2
   \]
   \[
   324 = 25 + MP^2
   \]
   \[
   MP^2 = 324 - 25 = 299
   \]
   \[
   MP = \sqrt{299} \approx 17.3 \, \text{cm}
   \]

4. Since \( M \) is the midpoint of \( PQ \), the length of the entire chord \( PQ \) is:
   \[
   PQ = 2 \times MP = 2 \times 17.3 \approx 34.6 \, \text{cm}
   \]

Thus, the length of the chord \( PQ \) is approximately **34.6 cm** (correct to 3 significant figures).

Would you like more details or have any questions? 

Here are some related questions:
1. What is the relationship between the radius and the distance from the center to the chord?
2. How does the Pythagorean theorem apply in finding the length of a chord?
3. Can the formula be applied if the perpendicular is not given but the angle subtended by the chord is?
4. What is the equation of a circle in Cartesian coordinates and how does it relate to this problem?
5. How do the properties of chords change if they are parallel?

**Tip:** Always remember that the perpendicular from the center of a circle to a chord bisects the chord, simplifying calculations.

In the diagram, O is the center of the circle. Find the length of the chord PQ correct to 3 s.f., where the radius of the circle is 18 cm, and the perpendicular distance from the center O to the chord PQ is 5 cm.

This problem involves finding the length of the chord PQ in a circle where the radius is 18 cm, and the perpendicular from the center to the chord is 5 cm. Using the Pythagorean theorem, we calculate the length of the chord. Suitable for students in Grades 9-10.

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