Question

graph the image of the given figure across the mirror line.

1. figure of a hexagon labeled e, d, c, b, a, f on a grid
2. figure of a square labeled t, u, v, w with a diagonal mirror line on a grid
3. figure of a rectangle labeled j, k, l, m with a vertical mirror line on a grid

Answer:

To solve the problem of graphing the image of a figure across a mirror line (reflection), we follow these general steps for each figure:

Step 1: Identify Key Points

For each vertex (or key point) of the original figure, locate its position on the coordinate grid. Let's denote a vertex as \( (x, y) \).

Step 2: Determine the Mirror Line

• For a horizontal mirror line (e.g., the x - axis in the first figure): The reflection of a point \( (x, y) \) across the horizontal line \( y = k \) (if the mirror line is \( y = k \)) is \( (x, 2k - y) \). If the mirror line is the x - axis (\( y = 0 \)), the reflection of \( (x, y) \) is \( (x, -y) \). In the first figure, if the mirror line is the horizontal line through the middle (let's assume it's \( y = 0 \) for simplicity, but we can also use the grid to count distances), for a point \( A(x_1,y_1) \), we count the number of units from \( A \) to the mirror line and then plot a point on the other side of the mirror line at the same distance.
• For a diagonal mirror line (e.g., \( y=x \) in the second figure): The reflection of a point \( (x, y) \) across the line \( y = x \) is \( (y, x) \). So if we have a point \( T(x_2,y_2) \), its reflection \( T' \) will be \( (y_2,x_2) \).
• For a vertical mirror line (e.g., the vertical line in the third figure): The reflection of a point \( (x, y) \) across the vertical line \( x = k \) is \( (2k - x, y) \). If the mirror line is a vertical line \( x = k \), we count the distance from the point \( (x,y) \) to \( x = k \) (i.e., \( |x - k| \)) and then plot a point at \( x=k - |x - k| \) (on the other side of the vertical line) with the same \( y \) - coordinate.

Step 3: Plot Reflected Points

For each vertex of the original figure, find its reflection using the appropriate rule for the mirror line and plot the reflected point.

Step 4: Connect Reflected Points

Once all the reflected vertices are plotted, connect them in the same order as the original figure to get the reflected image.

Example for Figure 1 (Horizontal Mirror Line)

1. Identify Vertices: Let's assume the coordinates of the vertices of the hexagon are \( A(x_A,y_A) \), \( B(x_B,y_B) \), \( C(x_C,y_C) \), \( D(x_D,y_D) \), \( E(x_E,y_E) \), \( F(x_F,y_F) \).
2. Reflect Each Vertex: Since the mirror line is horizontal, for a vertex \( (x,y) \), the reflected vertex \( (x, y') \) where \( y' \) is the distance from the mirror line on the opposite side. For example, if \( A \) is 2 units above the mirror line, the reflected \( A' \) will be 2 units below the mirror line with the same \( x \) - coordinate.
3. Plot and Connect: Plot all the reflected vertices \( A' \), \( B' \), \( C' \), \( D' \), \( E' \), \( F' \) and connect them to form the reflected hexagon.

Example for Figure 2 (Diagonal Mirror Line \( y = x \))

1. Identify Vertices: Let the vertices of the rectangle be \( T(x_T,y_T) \), \( U(x_U,y_U) \), \( V(x_V,y_V) \), \( W(x_W,y_W) \).
2. Reflect Each Vertex: Using the rule for reflection across \( y=x \), if \( T=(a,b) \), then \( T'=(b,a) \), if \( U=(c,d) \), then \( U'=(d,c) \), if \( V=(e,f) \), then \( V'=(f,e) \), if \( W=(g,h) \), then \( W'=(h,g) \).
3. Plot and Connect: Plot the reflected vertices \( T' \), \( U' \), \( V' \), \( W' \) and connect them to form the reflected rectangle.

Example for Figure 3 (Vertical Mirror Line)

1. Identify Vertices: Let the vertices of the rectangle be \( J(x_J,y_J) \), \( K(x_K,y_K) \), \( L(x_L,y_L) \), \( M(x_M,y_M) \).
2. Reflect Each Vertex: If the vertical mirror line is \( x = k \), for a vertex \( (x,y) \), the reflected vertex is \( (2k - x,y) \). For example, if \( J \) is 3 units to the right of the mirror line, the reflected \( J' \) will be 3 units to the left of the mirror line with the same \( y \) - coordinate.
3. Plot and Connect: Plot the reflected vertices \( J' \), \( K' \), \( L' \), \( M' \) and connect them to form the reflected rectangle.

Since the problem is about graphing reflections, the key is to use the properties of reflections (distance from the mirror line is preserved, and the mirror line is the perpendicular bisector of the segment joining a point and its reflection) to plot the reflected points and then connect them.

(Note: Since this is a graphing problem, the final answer would be the correctly graphed reflected figures. If we were to describe the process in terms of coordinates, we would follow the steps above for each figure's vertices.)