Question

rotate figure efgh 180° around the origin.

Explanation:

Step1: Identify coordinates of E, F, G, H

From the graph, let's assume the grid has each square as 1 unit. Let's find the coordinates:
- \( E \): Let's say \( E \) is at \( (1, 24) \)? Wait, no, looking at the grid, the origin is (0,0), and the y-axis goes up to 25, down to -25, x-axis from -25 to 25. Wait, actually, let's look at the positions. Let's see, E and H are on the same vertical line, F and G on the same vertical line. Let's assume the coordinates:
Wait, maybe better: Let's check the positions. Let's say E is at (1, 23)? No, maybe the coordinates are: Let's see, the distance from origin. Wait, maybe E is (1, 22)? No, perhaps the coordinates are: Let's look at the grid. Let's assume each square is 5 units? No, the grid lines are at 0, then each square is 5? Wait, no, the x and y axes have labels at -25, 0, 25. So each grid square is 5 units? Wait, no, the distance between -25 and 0 is 25, so if there are 5 grid lines (from -25 to 25, with 0 in the middle), each grid square is 5 units? Wait, no, the graph has a grid with squares, so from 0 to 25 on x-axis, how many squares? Let's count: from 0 to 25, there are 5 squares (since 25/5=5). So each square is 5 units. Wait, but the points E, F, G, H: Let's see, E is at (1, 24)? No, maybe the coordinates are: Let's look at the positions. Let's say E is (1, 23)? No, perhaps the coordinates are: Let's assume that E is (1, 22), H is (1, 2), F is (3, 22), G is (3, 2). Wait, no, that can't be. Wait, the y-axis goes up to 25, so 25 is the top. So maybe E is (1, 24), H is (1, 4), F is (3, 24), G is (3, 4). Wait, no, that's too high. Wait, maybe the coordinates are (1, 23), (3, 23), (3, 3), (1, 3). Wait, no, the distance from x-axis (y=0) to H and G is, say, 3 units up? No, the x-axis is at y=0, so H and G are on the line y=3? No, the graph shows H and G on the x-axis? Wait, no, the blue dots: E and F are on the top horizontal line, H and G on the bottom horizontal line. So E and H are vertical, F and G are vertical. So let's find the coordinates:

Wait, maybe the coordinates are:

- \( E \): Let's say \( E = (1, 22) \)
- \( H = (1, 2) \)
- \( F = (3, 22) \)
- \( G = (3, 2) \)

Wait, no, that's not right. Wait, the x-axis is at y=0, so H and G are on y=0? Wait, the blue dots: H and G are on the x-axis? Wait, the graph shows H and G on the x-axis (y=0), and E and F above them (y=24? No, the y-axis goes up to 25, so 24 is near the top. Wait, maybe the coordinates are:

- \( E = (1, 24) \)
- \( H = (1, 0) \)
- \( F = (3, 24) \)
- \( G = (3, 0) \)

Ah, that makes sense. So E is (1, 24), H is (1, 0), F is (3, 24), G is (3, 0). Wait, but the x-axis is at y=0, so H and G are on the x-axis (y=0), and E and F are on y=24 (top, near 25). So that's a rectangle with length 2 (from x=1 to x=3) and height 24 (from y=0 to y=24). Wait, no, the vertical distance from H (y=0) to E (y=24) is 24 units.

Now, to rotate a point \( (x, y) \) 180° around the origin, the rule is \( (x, y) ightarrow (-x, -y) \).

So let's find the coordinates of E, F, G, H:

- \( E \): Let's confirm. If H is (1, 0), E is (1, 24) (since they are vertical, same x, different y).
- \( F \): (3, 24) (same y as E, same x difference as G and H)
- \( G \): (3, 0) (same y as H, same x as F)
- \( H \): (1, 0)

Now, apply the 180° rotation rule: \( (x, y) ightarrow (-x, -y) \)

Step2: Rotate each point

- For \( E(1, 24) \): Rotated point \( E' \) is \( (-1, -24) \)
- For \( F(3, 24) \): Rotated point \( F' \) is \( (-3, -24) \)
- For \( G(3, 0) \): Rotated point \( G' \) is \( (-3, 0) \)
- For \( H(1, 0) \): Rotated point \( H' \) is \( (-1, 0) \)

Wait, but that seems odd. Wait, maybe my initial coordinates are wrong. Let's re-examine the graph. Maybe the coordinates are:

Wait, the x-axis and y-axis: the origin is (0,0). The points E, F, G, H: Let's look at the grid. Let's say each square is 5 units. So from 0 to 25 on x-axis, there are 5 squares (25/5=5), so each square is 5 units. So the x-coordinate for E: let's say E is at (5, 20), H at (5, 5), F at (15, 20), G at (15, 5). Wait, that makes more sense. Let's check:

- E: (5, 20)
- F: (15, 20)
- G: (15, 5)
- H: (5, 5)

Yes, that's a rectangle with length 10 (from x=5 to x=15) and height 15 (from y=5 to y=20). Now, rotating 180° around origin:

Rule: \( (x, y) ightarrow (-x, -y) \)

So:

- \( E(5, 20) ightarrow E'(-5, -20) \)
- \( F(15, 20) ightarrow F'(-15, -20) \)
- \( G(15, 5) ightarrow G'(-15, -5) \)
- \( H(5, 5) ightarrow H'(-5, -5) \)

Now, plot these points:

- \( E'(-5, -20) \)
- \( F'(-15, -20) \)
- \( G'(-15, -5) \)
- \( H'(-5, -5) \)

Now, connect these points to form the rotated figure.

Wait, maybe my initial coordinate assumption was wrong, but the key is the rule for 180° rotation: (x,y) becomes (-x,-y). So regardless of the initial coordinates, the process is:

1. Find the coordinates of each vertex of the figure.
2. Apply the 180° rotation transformation: (x, y) → (-x, -y)
3. Plot the new points and connect them to form the rotated figure.

So the steps are:

1. Identify coordinates of E, F, G, H.
2. Apply (x, y) → (-x, -y) to each.
3. Draw the new figure by connecting the rotated points.

Answer:

To rotate figure \( EFGH \) 180° around the origin, follow these steps:

Step 1: Identify Coordinates of Vertices

Assume the grid has 1-unit squares (or scale appropriately). From the graph:

• \( E(x_1, y_1) \), \( F(x_2, y_2) \), \( G(x_3, y_3) \), \( H(x_4, y_4) \) (e.g., if \( E = (1, 24) \), \( F = (3, 24) \), \( G = (3, 0) \), \( H = (1, 0) \) based on vertical/horizontal alignment).

Step 2: Apply 180° Rotation Rule

The transformation for a 180° rotation around the origin is:
\[ (x, y)
ightarrow (-x, -y) \]

• Rotate \( E(x_1, y_1) \): \( E'(-x_1, -y_1) \)
• Rotate \( F(x_2, y_2) \): \( F'(-x_2, -y_2) \)
• Rotate \( G(x_3, y_3) \): \( G'(-x_3, -y_3) \)
• Rotate \( H(x_4, y_4) \): \( H'(-x_4, -y_4) \)

Step 3: Plot and Connect Rotated Vertices

Plot the new points \( E', F', G', H' \) and connect them to form the rotated figure.

For example, if \( E = (1, 24) \), \( F = (3, 24) \), \( G = (3, 0) \), \( H = (1, 0) \):

• \( E' = (-1, -24) \)
• \( F' = (-3, -24) \)
• \( G' = (-3, 0) \)
• \( H' = (-1, 0) \)

Connect \( E' \to F' \to G' \to H' \to E' \) to complete the rotated figure.

(Note: Adjust coordinates based on the actual grid scale; the key is applying \( (x, y) \to (-x, -y) \) to each vertex.)