This question involves the properties of intersecting chords that meet outside the circle at a common point. When two chords intersect outside a circle, the following relationship holds:

$AP \times AB = CP \times CD$

Given values:

• $AP = 200$ mm
• $AB = 120$ mm
• $CP = 160$ mm
• $CD = ?$ (we need to find this)

Solution

Using the intersecting chords theorem:

$AP \times AB = CP \times CD$

Substituting the given values:

$200 \times 120 = 160 \times CD$

Calculating the left side:

$200 \times 120 = 24000$

So,

$24000 = 160 \times CD$

Now, divide both sides by 160 to solve for $CD$:

$CD = \frac{24000}{160} = 150 \text{ mm}$

Answer

The length of $CD$ is 150 mm.

Would you like further explanation, or do you have any questions?

Here are five related questions to deepen your understanding:

1. What is the intersecting chords theorem and how is it applied in different scenarios?
2. How would the answer change if $AP$ were different?
3. Can this formula be applied if the chords intersect inside the circle? Why or why not?
4. What other theorems are used with circles and intersecting lines?
5. How can you verify your answer using an alternative approach?

Tip: Always write down each step of the calculation carefully to avoid simple arithmetic errors, especially when dealing with large numbers.