Let's solve each problem step by step.

Problem 9: Reflection across the line $x = 3$

We are tasked with finding the image of point $(8, -7)$ across the vertical line $x = 3$.

1. Understand the reflection rule:
   When reflecting across a vertical line $x = c$, the distance from the original point to the line is the same as the distance from the reflected point to the line, but on the opposite side.
   Only the $x$-coordinate changes, while the $y$-coordinate remains the same.

2. Steps to calculate:

   • The original point is $(8, -7)$.
   • The line $x = 3$ is the mirror.
   • Calculate the distance between $x = 8$ and $x = 3$:
     $\text{Distance} = 8 - 3 = 5$
   • The reflected point will be $5$ units to the left of $x = 3$:
     $\text{Reflected } x = 3 - 5 = -2$
   • The $y$-coordinate remains unchanged, so the reflected point is:
     $(-2, -7)$

3. Answer:
   The correct option is (4) $(-2, -7)$.

Problem 10: Reflection of $(3, 9)$ to $(7, 9)$

We need to determine the equation of the line of reflection.

1. Understand the scenario:

   • The original point is $(3, 9)$.
   • The reflected point is $(7, 9)$.
   • The $y$-coordinate remains constant at $y = 9$, which means the reflection happens horizontally.

2. Midpoint of the reflection:
   The line of reflection is the vertical line passing through the midpoint of $(3, 9)$ and $(7, 9)$:

   • Midpoint formula:
     $\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
   • Substituting $x_1 = 3$, $x_2 = 7$, and $y_1 = y_2 = 9$:
     $\text{Midpoint} = \left( \frac{3 + 7}{2}, 9 \right) = (5, 9)$
   • Therefore, the line of reflection is the vertical line $x = 5$.

3. Answer:
   The correct option is (4) $x = 5$.

Summary of Answers:

• Question 9: (4) $(-2, -7)$
• Question 10: (4) $x = 5$

Do you need further clarification or a detailed explanation of any step? Let me know!

Related Questions:

1. What is the general formula for reflecting a point across a vertical line $x = c$?
2. How do you find the line of reflection given two symmetric points?
3. What happens when a point is reflected across the $x$-axis or $y$-axis?
4. How would you reflect a point across a diagonal line like $y = x$?
5. How is the midpoint formula applied to find the line of symmetry?

Tip:

When reflecting points, always focus on which coordinate (x or y) changes and calculate the distance to the mirror line carefully.