Verified Answer :
- Identify the vertices of the trapezoid and the center of dilation \(A(-2, 3)\) with scale factor \(k=2\).
- Apply the dilation formula \(P'(x', y') = (x_0 + k(x - x_0), y_0 + k(y - y_0))\) to each vertex.
- Calculate the resulting coordinates for the vertices: \(A'(-2, 3)\), \(B'(6, 3)\), \(C'(12, -7)\), and \(D'(-2, -7)\).
- Conclude that the image of the trapezoid is defined by these new vertices: \(\boxed{A'(-2, 3), B'(6, 3), C'(12, -7), D'(-2, -7)}\).
Explanation
- Analyze the transformation and dilation formula.
The trapezoid has vertices \(A(-2, 3)\), \(B(2, 3)\), \(C(5, -2)\), and \(D(-2, -2)\). We are performing a dilation with respect to vertex \(A(-2, 3)\) using a scale factor \(k = 2\).
The formula for dilating a point \(P(x, y)\) with respect to a center of dilation \(O(x_0, y_0)\) by a scale factor \(k\) is:
\[P'(x', y') = (x_0 + k(x - x_0), y_0 + k(y - y_0))\]
In this case, the center of dilation is \(A(-2, 3)\), so \((x_0, y_0) = (-2, 3)\). We apply this formula to each of the vertices.
- Calculate the coordinates of the image vertices.
Applying the dilation formula \(P'(x', y') = (-2 + 2(x - (-2)), 3 + 2(y - 3))\):
- For vertex \(A(-2, 3)\):
Since \(A\) is the center of dilation, it remains unchanged:
\(A' = (-2 + 2(-2 - (-2)), 3 + 2(3 - 3)) = (-2 + 2(0), 3 + 2(0)) = (-2, 3)\).
- For vertex \(B(2, 3)\):
\(B' = (-2 + 2(2 - (-2)), 3 + 2(3 - 3)) = (-2 + 2(4), 3 + 2(0)) = (-2 + 8, 3 + 0) = (6, 3)\).
- For vertex \(C(5, -2)\):
\(C' = (-2 + 2(5 - (-2)), 3 + 2(-2 - 3)) = (-2 + 2(7), 3 + 2(-5)) = (-2 + 14, 3 - 10) = (12, -7)\).
- For vertex \(D(-2, -2)\):
\(D' = (-2 + 2(-2 - (-2)), 3 + 2(-2 - 3)) = (-2 + 2(0), 3 + 2(-5)) = (-2 + 0, 3 - 10) = (-2, -7)\).
- Conclusion of the transformed coordinates.
After applying the dilation with scale factor \(k = 2\) centered at vertex \(A(-2, 3)\), the coordinates of the image vertices are:
\(A' = (-2, 3)\)
\(B' = (6, 3)\)
\(C' = (12, -7)\)
\(D' = (-2, -7)\)
The final answer is \(\boxed{A'(-2, 3), B'(6, 3), C'(12, -7), D'(-2, -7)}\).
Examples
Dilation is commonly used in graphic design and digital image editing when scaling an image relative to a specific point (like a corner or center). For example, if you are designing a logo and want to resize a geometric shape proportionally from its top-left corner, you are essentially performing a mathematical dilation. By calculating the new coordinates using this method, graphic software ensures that the shape maintains its proportions and integrity while expanding or shrinking to fit your layout.