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The question asks to find the length of \( PQ \), the distance between the points \( P \) and \( Q \) on two circles with known radii. Here’s a breakdown of the information we have:

1. **Circle Q**: has a radius \( r = 2.1 \, \text{cm} \).
2. **Circle P**: has a radius \( r = 1.2 \, \text{cm} \).
3. **Another circle**: has a radius \( r = 1.4 \, \text{cm} \) and is tangent to both circles at some points.
4. **Angle**: \( \angle PQ = 106^\circ \).

To find the distance \( PQ \) between the centers of Circle Q and the circle containing point P, we can use the formula for the distance between two points given the angle between two tangent points. This will involve using trigonometric relationships and the Law of Cosines.

Let’s calculate it using the given values:

The distance between \( P \) and \( Q \), which we’ll call \( d \), can be found using the Law of Cosines formula:

\[
d = \sqrt{(r_P + r_Q)^2 - 2 \cdot r_P \cdot r_Q \cdot \cos(\angle PQ)}
\]

Here:
- \( r_P = 1.2 \, \text{cm} \)
- \( r_Q = 2.1 \, \text{cm} \)
- \( \angle PQ = 106^\circ \)

Let’s substitute these values and solve:

\[
PQ = \sqrt{(1.2 + 2.1)^2 - 2 \cdot 1.2 \cdot 2.1 \cdot \cos(106^\circ)}
\]

Let me know if you’d like me to calculate this expression for you. 

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Would you like further details or have any questions?

Here are 5 related questions:
1. What is the Law of Cosines, and how is it used in geometry?
2. How do we calculate angles in radians and degrees when solving trigonometric problems?
3. Why do we use the Law of Cosines instead of the Pythagorean theorem in this problem?
4. How does the angle between two lines affect the distance between points?
5. What are the trigonometric values of common angles like 106 degrees?

**Tip:** Always make sure your calculator is set to the correct angle mode (degrees/radians) when working with trigonometric functions.

Three circles are arranged as shown in the figure. Find the length of PQ.

This problem involves using the Law of Cosines to find the distance between two points on different circles with known radii. Suitable for high school students, it reinforces concepts in geometry and trigonometry.

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