Question

question the figure below is dilated by a factor of 3 centered at the origin. plot the resulting image. click twice to plot a segment. click a segment to delete it.

Explanation:

To solve the dilation of a figure by a factor of 3 centered at the origin, we follow these steps:

Step 1: Identify the coordinates of the vertices of the original figure.

Let's assume the original vertices (from the graph) are:

• Let's denote the vertices as \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), etc. From the graph, let's pick some key points (assuming the small triangle and the larger triangle, but focusing on the figure to dilate). Wait, actually, the figure has vertices. Let's look at the coordinates:

Looking at the graph, let's identify the vertices of the figure (the polygon). Let's say the original vertices are:

• Vertex 1: Let's say (0, 9) [top on y-axis]
• Vertex 2: (-5, 3) [left point]
• Vertex 3: (0, -9) [bottom on y-axis]
• Vertex 4: (2, 0) [right point] Wait, maybe better to take each vertex and apply dilation.

Dilation rule: For a point \( (x, y) \) dilated by a factor \( k \) centered at the origin, the new point is \( (k \cdot x, k \cdot y) \).

So let's list the original vertices (from the graph, let's assume the figure has vertices at:

1. (0, 9)
2. (-5, 3)
3. (0, -9)
4. (2, 0) Wait, maybe the figure is a polygon with vertices. Let's check the small triangle and the larger one. Wait, the problem says "the figure below" – let's assume the figure has vertices, say, let's take each vertex:

Wait, maybe the original figure has vertices:

• Let's take the top vertex: (0, 9)
• Left vertex: (-5, 3)
• Bottom vertex: (0, -9)
• Right vertex: (2, 0) Wait, no, maybe the figure is a quadrilateral? Wait, the graph shows a figure with vertices. Let's correctly identify the coordinates. Let's look at the grid:

Assuming each grid square is 1 unit. Let's take the vertices:

1. Top: (0, 9) (since it's on y-axis, 9 units up)
2. Left: (-5, 3) (5 units left on x-axis, 3 units up on y-axis)
3. Bottom: (0, -9) (9 units down on y-axis)
4. Right: (2, 0) (2 units right on x-axis, 0 on y-axis) Wait, no, maybe the figure is a triangle? Wait, the problem says "the figure" – maybe it's a polygon with vertices. Let's proceed with dilation.

Step 2: Apply the dilation factor of 3 to each vertex.

For a dilation centered at the origin with factor \( k = 3 \), the transformation is \( (x, y) \to (3x, 3y) \).

Let's take each vertex:

1. Vertex 1: (0, 9) → \( (3 \cdot 0, 3 \cdot 9) = (0, 27) \)
2. Vertex 2: (-5, 3) → \( (3 \cdot (-5), 3 \cdot 3) = (-15, 9) \)
3. Vertex 3: (0, -9) → \( (3 \cdot 0, 3 \cdot (-9)) = (0, -27) \)
4. Vertex 4: (2, 0) → \( (3 \cdot 2, 3 \cdot 0) = (6, 0) \)

Wait, but maybe the original figure has different vertices. Let's check the small triangle (V, T, H). Let's say the small triangle has vertices:

• V: (-1, 1)
• T: (0, -2)
• H: (2, 0)

Then dilating by 3:

• V: \( (3 \cdot (-1), 3 \cdot 1) = (-3, 3) \)
• T: \( (3 \cdot 0, 3 \cdot (-2)) = (0, -6) \)
• H: \( (3 \cdot 2, 3 \cdot 0) = (6, 0) \)

But the larger figure:

• Top: (0, 9) → (0, 27)
• Left: (-5, 3) → (-15, 9)
• Bottom: (0, -9) → (0, -27)
• Right: (2, 0) → (6, 0)

Wait, maybe the figure is a polygon with vertices at (0,9), (-5,3), (0,-9), (2,0). So after dilation:

• (0,9) → (0,27)
• (-5,3) → (-15,9)
• (0,-9) → (0,-27)
• (2,0) → (6,0)

Then we plot these new points and connect them to form the dilated figure.

Step 3: Plot the new vertices.

Plot the points (0, 27), (-15, 9), (0, -27), and (6, 0) (or the correct vertices based on the original figure) and connect them to get the dilated image.

Answer:

To plot the dilated figure, follow these steps:

1. Identify Original Vertices: Determine the coordinates of each vertex of the original figure from the graph.
2. Apply Dilation: For each vertex \((x, y)\), compute the new coordinates as \((3x, 3y)\) (since the dilation factor is 3 and centered at the origin).
3. Plot New Vertices: Mark the new coordinates on the grid and connect them to form the dilated image.

For example, if the original vertices are \((0, 9)\), \((-5, 3)\), \((0, -9)\), and \((2, 0)\), the dilated vertices are \((0, 27)\), \((-15, 9)\), \((0, -27)\), and \((6, 0)\). Plot these points and connect them to complete the dilation.