https://file.aiquickdraw.com/mathbot/file/1730544587245xbsihb8p.jpg

To find the length of \( PQ \) in this diagram, let’s analyze the geometry provided. Here’s the information from the diagram:

1. Circle at \( Q \) has a radius of \( r = 2.5 \, \text{cm} \).
2. Circle at \( P \) has a radius of \( r = 1.3 \, \text{cm} \).
3. A third circle with radius \( r = 1.6 \, \text{cm} \) is also involved in the arrangement.
4. The angle between the line segments connecting the centers \( P \) and \( Q \) and the third circle's center is \( 106^\circ \).

Since we are asked to find the length \( PQ \), we’ll apply the Law of Cosines to solve this.

### Steps:

1. **Identify the distances and angle for the Law of Cosines**:
   - \( PQ \) is the distance between the centers of the circles with radii \( 2.5 \, \text{cm} \) and \( 1.6 \, \text{cm} \).
   - The angle between the line from \( P \) to \( Q \) is \( 106^\circ \).

2. **Apply the Law of Cosines**:
   The Law of Cosines states:
   \[
   PQ^2 = (r_Q)^2 + (r_P)^2 - 2 \cdot r_Q \cdot r_P \cdot \cos(\theta)
   \]
   where:
   - \( r_Q = 2.5 \, \text{cm} \),
   - \( r_P = 1.6 \, \text{cm} \), and
   - \( \theta = 106^\circ \).

3. **Calculate**:
   Substitute the values into the equation and solve for \( PQ \).

Let's perform the calculation:

\[
PQ^2 = (2.5)^2 + (1.6)^2 - 2 \cdot 2.5 \cdot 1.6 \cdot \cos(106^\circ)
\]

1. **Square the radii**:
   - \( (2.5)^2 = 6.25 \)
   - \( (1.6)^2 = 2.56 \)

2. **Calculate the cosine term**:
   - \( \cos(106^\circ) \approx -0.276 \)

3. Substitute:
   \[
   PQ^2 = 6.25 + 2.56 - 2 \cdot 2.5 \cdot 1.6 \cdot (-0.276)
   \]
4. Simplify and solve for \( PQ \):

I’ll complete the calculation for the exact length, rounding to one decimal as requested.The length of \( PQ \) is approximately \( 3.3 \, \text{cm} \).

Would you like more details on the steps involved or have any questions?

Here are five related questions to deepen your understanding:

1. How does the Law of Cosines apply to triangles with obtuse angles?
2. What would change if the angle between \( P \) and \( Q \) were different?
3. How would you find the distance between centers if only radii and area overlap were provided?
4. How is the Law of Cosines derived, and why is it useful for non-right triangles?
5. Could this problem be solved using vector components?

**Tip:** When solving geometry problems involving circles and distances, visualize the setup and check if trigonometric laws like Sine or Cosine Laws could simplify the process.

Three circles are arranged as shown in the figure. Find the length of PQ, where the radius of circle at Q is 2.5 cm, the radius of circle at P is 1.3 cm, the radius of a third circle is 1.6 cm, and the angle between the line segments connecting the centers P and Q and the third circle's center is 106°.

Solve for the length of PQ in a geometry problem involving three circles with given radii and an angle between centers. This problem applies the Law of Cosines, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation