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The problem asks for the distance from point \( B \) to focus \( F_2 \), given that the distance from \( B \) to focus \( F_1 \) is 1.5 units.

This setup looks like a hyperbola, where points \( F_1 \) and \( F_2 \) are the foci. A fundamental property of hyperbolas is that the difference in distances from any point on the hyperbola to the two foci is constant. The equation governing this property is:

\[
|d(B, F_1) - d(B, F_2)| = 2a
\]

where \( 2a \) is the distance between the two vertices \( V_1 \) and \( V_2 \) on the hyperbola.

### Steps to solve:
1. **Measure the vertices' distance \( V_1 \) to \( V_2 \)**: Using the graph, the coordinates of \( V_1 \) and \( V_2 \) are \( (-4, 0) \) and \( (4, 0) \), so:
   \[
   2a = 4 - (-4) = 8 \implies a = 4
   \]
   Therefore, \( |d(B, F_1) - d(B, F_2)| = 8 \).

2. **Use the given distance from \( B \) to \( F_1 \)**: The distance from \( B \) to \( F_1 \) is 1.5 units.

Let \( d(B, F_2) \) be the unknown distance from \( B \) to \( F_2 \). According to the hyperbola property:

\[
|1.5 - d(B, F_2)| = 8
\]

3. **Solve for \( d(B, F_2) \)**:

   \[
   1.5 - d(B, F_2) = -8 \quad \text{(since it must be negative to satisfy the equation)}
   \]
   \[
   d(B, F_2) = 1.5 + 8 = 9.5 \, \text{units}
   \]

Thus, the distance from point \( B \) to focus \( F_2 \) is **9.5 units**.

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Would you like more details on this solution or have any questions?

Here are 5 related questions:
1. How do we derive the equation of a hyperbola given its foci and vertices?
2. What are the differences between an ellipse and a hyperbola in terms of focal properties?
3. How do you calculate distances between points on a coordinate plane?
4. What role does the transverse axis play in defining the shape of a hyperbola?
5. How does the eccentricity of a hyperbola influence its geometry?

**Tip:** In a hyperbola, the foci are always farther from the center than the vertices, and the difference in distances from any point on the curve to the foci is constant!

If point B is 1.5 units away from F1, how far is point B from F2?

In this math problem, we explore how to find the distance from a point on a hyperbola to one of its foci, given the distance to the other focus. Using the focal property of hyperbolas and key geometric principles, this problem provides a detailed solution for students in Grades 10-12.

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