Question

1. reflect the triangle across the diagonal line $y = x$ and draw the image. then label the image with $m$, $n$, $o$.

Explanation:

Step1: Recall Reflection over \( y = x \)

To reflect a point \((a, b)\) over the line \( y = x \), we swap the \( x \)- and \( y \)-coordinates. So the image of \((a, b)\) is \((b, a)\).

Step2: Identify Original Points

First, determine the coordinates of \( M \), \( N \), and \( O \) from the graph. Let's assume (from the grid) the coordinates:

• Let \( M = (x_1, y_1) \), \( N = (x_2, y_2) \), \( O = (x_3, y_3) \). For example, if \( O \) is at \((0, 1)\) (from the graph's vertical and horizontal grid), \( M \) at \((-2, -1)\), \( N \) at \((2, -2)\) (these are example coordinates; actual coordinates depend on the grid).

Step3: Apply Reflection

For each point:

• Reflect \( M(x_1, y_1) \): \( M' = (y_1, x_1) \)
• Reflect \( N(x_2, y_2) \): \( N' = (y_2, x_2) \)
• Reflect \( O(x_3, y_3) \): \( O' = (y_3, x_3) \)

Step4: Draw the Image

Plot the reflected points \( M' \), \( N' \), \( O' \) on the coordinate plane and connect them to form the triangle. Label them as \( M' \), \( N' \), \( O' \).

Answer:

To solve, follow these steps:

1. Find Coordinates: Determine \((x, y)\) for \( M \), \( N \), \( O \) from the grid.
2. Reflect Over \( y = x \): Swap \( x \) and \( y \) for each point (e.g., \((a, b) \to (b, a)\)).
3. Plot & Label: Draw the triangle using \( M' \), \( N' \), \( O' \) and label the vertices.

(Note: The actual drawing requires using the grid to plot the reflected points. For example, if \( O = (0, 1) \), \( O' = (1, 0) \); if \( M = (-2, -1) \), \( M' = (-1, -2) \); if \( N = (2, -2) \), \( N' = (-2, 2) \). Then connect these points.)