Question

polygon abcd is shown in the coordinate grid. match the dilations of abcd to the images showing the result of the dilation.

Explanation:

To solve the problem of matching dilations of polygon \(ABCD\) to their images, we first identify the coordinates of the vertices of \(ABCD\):

• \(A(1, 2)\)
• \(B(3, 4)\)
• \(C(6, 4)\)
• \(D(6, 2)\)

Key Concept: Dilation

A dilation is a transformation that scales a figure by a scale factor (\(k\)) relative to a center of dilation (usually the origin, \((0,0)\), unless stated otherwise). The new coordinates of a point \((x, y)\) after dilation with scale factor \(k\) are \((kx, ky)\).

Step 1: Analyze the Original Polygon

First, determine the lengths of sides or the “size” of \(ABCD\):

• Horizontal side \(AD\): From \(x=1\) to \(x=6\) (length \(6 - 1 = 5\)) or from \(A(1,2)\) to \(D(6,2)\) (length \(5\)).
• Vertical side \(CD\): From \(y=2\) to \(y=4\) (length \(4 - 2 = 2\)) or from \(C(6,4)\) to \(D(6,2)\) (length \(2\)).

Step 2: Predict Dilations

For each dilation (e.g., scale factor \(k = 0.5\), \(k = 2\), etc.):

• Scale factor \(k > 1\): The image will be larger than \(ABCD\).
• Scale factor \(0 < k < 1\): The image will be smaller than \(ABCD\).
• Scale factor \(k = 1\): The image is congruent (same size).

Example: Dilation with \(k = 2\) (Center at Origin)

New coordinates:

• \(A(1,2) \to (1 \cdot 2, 2 \cdot 2) = (2, 4)\)
• \(B(3,4) \to (3 \cdot 2, 4 \cdot 2) = (6, 8)\)
• \(C(6,4) \to (6 \cdot 2, 4 \cdot 2) = (12, 8)\)
• \(D(6,2) \to (6 \cdot 2, 2 \cdot 2) = (12, 4)\)

This image will be twice as large as \(ABCD\).

Example: Dilation with \(k = 0.5\) (Center at Origin)

New coordinates:

• \(A(1,2) \to (1 \cdot 0.5, 2 \cdot 0.5) = (0.5, 1)\)
• \(B(3,4) \to (3 \cdot 0.5, 4 \cdot 0.5) = (1.5, 2)\)
• \(C(6,4) \to (6 \cdot 0.5, 4 \cdot 0.5) = (3, 2)\)
• \(D(6,2) \to (6 \cdot 0.5, 2 \cdot 0.5) = (3, 1)\)

This image will be half the size of \(ABCD\).

Step 3: Match to Images

To match, compare the size (and coordinates) of each image to the predicted dilation:

• If an image is smaller, it has a scale factor \(0 < k < 1\).
• If an image is larger, it has a scale factor \(k > 1\).

Final Approach

1. Identify the center of dilation (assume origin unless given).
2. Calculate new coordinates for each vertex using \( (kx, ky) \).
3. Compare the shape and size of the image to the calculated coordinates.

For example, if one image has vertices at \((2, 4)\), \((6, 8)\), \((12, 8)\), \((12, 4)\), it matches the dilation with \(k = 2\). If another image has vertices at \((0.5, 1)\), \((1.5, 2)\), \((3, 2)\), \((3, 1)\), it matches \(k = 0.5\).

To complete the matching, apply this process to each dilation option and its corresponding image!

Answer:

To solve the problem of matching dilations of polygon \(ABCD\) to their images, we first identify the coordinates of the vertices of \(ABCD\):

• \(A(1, 2)\)
• \(B(3, 4)\)
• \(C(6, 4)\)
• \(D(6, 2)\)

Key Concept: Dilation

A dilation is a transformation that scales a figure by a scale factor (\(k\)) relative to a center of dilation (usually the origin, \((0,0)\), unless stated otherwise). The new coordinates of a point \((x, y)\) after dilation with scale factor \(k\) are \((kx, ky)\).

Step 1: Analyze the Original Polygon

First, determine the lengths of sides or the “size” of \(ABCD\):

• Horizontal side \(AD\): From \(x=1\) to \(x=6\) (length \(6 - 1 = 5\)) or from \(A(1,2)\) to \(D(6,2)\) (length \(5\)).
• Vertical side \(CD\): From \(y=2\) to \(y=4\) (length \(4 - 2 = 2\)) or from \(C(6,4)\) to \(D(6,2)\) (length \(2\)).

Step 2: Predict Dilations

For each dilation (e.g., scale factor \(k = 0.5\), \(k = 2\), etc.):

• Scale factor \(k > 1\): The image will be larger than \(ABCD\).
• Scale factor \(0 < k < 1\): The image will be smaller than \(ABCD\).
• Scale factor \(k = 1\): The image is congruent (same size).

Example: Dilation with \(k = 2\) (Center at Origin)

New coordinates:

• \(A(1,2) \to (1 \cdot 2, 2 \cdot 2) = (2, 4)\)
• \(B(3,4) \to (3 \cdot 2, 4 \cdot 2) = (6, 8)\)
• \(C(6,4) \to (6 \cdot 2, 4 \cdot 2) = (12, 8)\)
• \(D(6,2) \to (6 \cdot 2, 2 \cdot 2) = (12, 4)\)

This image will be twice as large as \(ABCD\).

Example: Dilation with \(k = 0.5\) (Center at Origin)

New coordinates:

• \(A(1,2) \to (1 \cdot 0.5, 2 \cdot 0.5) = (0.5, 1)\)
• \(B(3,4) \to (3 \cdot 0.5, 4 \cdot 0.5) = (1.5, 2)\)
• \(C(6,4) \to (6 \cdot 0.5, 4 \cdot 0.5) = (3, 2)\)
• \(D(6,2) \to (6 \cdot 0.5, 2 \cdot 0.5) = (3, 1)\)

This image will be half the size of \(ABCD\).

Step 3: Match to Images

To match, compare the size (and coordinates) of each image to the predicted dilation:

• If an image is smaller, it has a scale factor \(0 < k < 1\).
• If an image is larger, it has a scale factor \(k > 1\).

Final Approach

1. Identify the center of dilation (assume origin unless given).
2. Calculate new coordinates for each vertex using \( (kx, ky) \).
3. Compare the shape and size of the image to the calculated coordinates.

For example, if one image has vertices at \((2, 4)\), \((6, 8)\), \((12, 8)\), \((12, 4)\), it matches the dilation with \(k = 2\). If another image has vertices at \((0.5, 1)\), \((1.5, 2)\), \((3, 2)\), \((3, 1)\), it matches \(k = 0.5\).

To complete the matching, apply this process to each dilation option and its corresponding image!