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To solve this problem, let's analyze the given information in the diagram:

1. **Circle Radii**:
   - The radius of the first circle is \( r = 1.9 \) cm.
   - The radius of the second circle is \( r = 1.8 \) cm.
   - The radius of the third circle is \( r = 1.4 \) cm.

2. **Angle Measurement**:
   - The angle between the centers of the first and second circles with respect to the line segment \( PQ \) is \( 114^\circ \).

3. **Goal**:
   - We need to find the distance \( PQ \), which is the distance between the centers of the circles with radii 1.9 cm and 1.4 cm.

### Step-by-Step Solution

1. **Determine the Geometry of \( PQ \)**:
   - We have three circles arranged in such a way that the centers of the first and third circles form a line with an angle \( 114^\circ \) at point \( Q \) (the center of the second circle).
   - Using the Law of Cosines on triangle \( \Delta PQC \), where:
     - \( QC = 1.9 + 1.8 = 3.7 \) cm (distance between centers of circles with radii 1.9 cm and 1.8 cm),
     - \( PC = 1.4 + 1.8 = 3.2 \) cm (distance between centers of circles with radii 1.4 cm and 1.8 cm),
     - \( \angle PQC = 114^\circ \).

2. **Apply the Law of Cosines**:
   The Law of Cosines states that for any triangle with sides \( a \), \( b \), and \( c \) and opposite angle \( \theta \):
   \[
   c^2 = a^2 + b^2 - 2ab \cos(\theta)
   \]
   where \( c \) is the length of the side opposite the angle \( \theta \).

   Here:
   - Let \( PQ = c \),
   - \( QC = 3.7 \) cm,
   - \( PC = 3.2 \) cm,
   - \( \angle PQC = 114^\circ \).

   Substituting these values:
   \[
   PQ^2 = 3.7^2 + 3.2^2 - 2 \cdot 3.7 \cdot 3.2 \cdot \cos(114^\circ)
   \]

3. **Calculate \( PQ \)**:
   - First, calculate each part separately:
     - \( 3.7^2 = 13.69 \),
     - \( 3.2^2 = 10.24 \),
     - \( \cos(114^\circ) \approx -0.4067 \).

   Substitute and simplify:
   \[
   PQ^2 = 13.69 + 10.24 + 2 \cdot 3.7 \cdot 3.2 \cdot 0.4067
   \]
   \[
   PQ^2 \approx 13.69 + 10.24 + 2 \cdot

Three circles are arranged as shown in the figure on the right. Find the length of PQ. The radius of each circle is given as follows: r = 1.9 cm, r = 1.8 cm, and r = 1.4 cm, with an angle of 114° between the centers of two circles. What is the length of PQ?

Solve for the distance PQ between the centers of circles with radii 1.9 cm, 1.8 cm, and 1.4 cm, and an angle of 114° between them. This geometry problem demonstrates the use of the Law of Cosines, a fundamental trigonometric formula.

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