From the image, we can solve the problem step by step:

Observing the lines and slopes:

1. The dotted lines represent lines that are diagonals of the coordinate system, crossing through the points of interest. To calculate the slope, we pick two points on one of the dotted lines.

Step 1: Find the slope of one of the dotted lines:

For the first dotted line, let’s take two points:

• Point 1: (0, 0) (intersection at origin)
• Point 2: (4, 4)

The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{4 - 0} = 1$

The slope of one of the dotted lines is $1$.

Step 2: Slope of the line of reflection:

The line of reflection in the image seems to cross through the points:

• Point 1: (0, 3)
• Point 2: (3, 0)

The slope $m$ is: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{3 - 0} = -1$

The slope of the line of reflection is $-1$.

Final Answer:

• Slope of one of the dotted lines: $1$
• Slope of the line of reflection: $-1$

Would you like me to explain any steps further or provide additional clarification?

Additional Related Questions:

1. How do we determine the line of reflection algebraically?
2. Can you explain how to reflect a triangle across a line in coordinate geometry?
3. Why does the slope $-1$ indicate a perpendicular line to a slope of $1$?
4. What are other ways to find the line of reflection geometrically?
5. How do transformations preserve the shape and orientation of the triangle?

Tip:

Always check the coordinates of points carefully when calculating slopes; even small errors in coordinates can lead to incorrect slopes.