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Circle Ω has diameter AB = 5. Circles ω1 and ω2 have radii 1 and 2 respectively,
and are internally tangent to Ω at A and B respectively. The common chord of
ω1 and ω2 intersects Ω at P and Q. What is the length of PQ?

To solve this problem, we need to analyze the relationship between the given circles and the chord \( PQ \) that intersects the circle \( \Omega \).

### Step 1: Understanding the Geometry
- \( \Omega \) is the large circle with a diameter \( AB = 5 \).
- \( \omega_1 \) and \( \omega_2 \) are smaller circles internally tangent to \( \Omega \) at points \( A \) and \( B \) respectively.
  - \( \omega_1 \) has a radius \( r_1 = 1 \).
  - \( \omega_2 \) has a radius \( r_2 = 2 \).
- \( P \) and \( Q \) are the points where the common chord of \( \omega_1 \) and \( \omega_2 \) intersects \( \Omega \).

### Step 2: Using Power of a Point Theorem
The key to solving this problem is applying the power of a point theorem. The power of a point \( P \) with respect to a circle is given by \( |PA \cdot PB| \) for any point \( P \) outside the circle.

### Step 3: Coordinate System Setup
Let's set up a coordinate system:
- Place \( A \) at \( (-\frac{5}{2}, 0) \) and \( B \) at \( (\frac{5}{2}, 0) \).
- The center \( O \) of \( \Omega \) is at the origin \( (0, 0) \).

For the smaller circles:
- The center of \( \omega_1 \) is at \( A_1 \) located at \( (-\frac{3}{2}, 0) \) with a radius of 1.
- The center of \( \omega_2 \) is at \( A_2 \) located at \( (\frac{1}{2}, 0) \) with a radius of 2.

### Step 4: Distance Between Centers of \( \omega_1 \) and \( \omega_2 \)
The distance \( d \) between the centers \( A_1 \) and \( A_2 \) is:
\[
d = |A_1 A_2| = \left|\frac{1}{2} - \left(-\frac{3}{2}\right)\right| = 2
\]

### Step 5: Length of Common Chord PQ
To find the length of the common chord \( PQ \), we use the fact that the length of the common chord between two intersecting circles is given by:
\[
PQ = 2 \sqrt{r_1^2 - \left(\frac{d^2 + r_1^2 - r_2^2}{2d}\right)^2}
\]
Substituting the values:
\[
PQ = 2 \sqrt{1^2 - \left(\frac{2^2 + 1^2 - 2^2}{2 \cdot 2}\right)^2}
\]
Simplifying further:
\[
PQ = 2 \sqrt{1 - \left(\frac{4 + 1 - 4}{4}\right)^2} = 2 \sqrt{1 - \left(\frac{1}{4}\right)^2} = 2 \sqrt{1 - \frac{1}{16}} = 2 \sqrt{\frac{15}{16}} = \frac{2 \cdot \sqrt{15}}{4} = \frac{\sqrt{15}}{2}
\]

### Final Answer
The length of \( PQ \) is \( \sqrt{15} \).

Let me know if you want more details or have any questions. Here are some related questions you might be interested in:

1. How does the power of a point theorem apply to this situation?
2. What are the properties of the common chord of two intersecting circles?
3. How can the equation of a circle be used to find the length of a chord?
4. What happens to the common chord's length if the radii of the circles change?
5. How does the internal tangency of circles affect the geometry of the system?

**Tip:** When dealing with circle problems, always consider the power of a point theorem as it is a powerful tool for finding distances and lengths related to chords and tangents.

Explore how to calculate the length of common chord PQ in a circle geometry problem involving intersecting circles and a larger circle. Detailed step-by-step solution using the Power of a Point Theorem. Suitable for advanced high school students.

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