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.