Use the image to answer the question.

To solve the questions about the dilation of the parallelograms and the triangle, let's break them down one at a time.

1. Dilation of Parallelogram ABCD to A'B'C'D': Since the specific coordinates of the points A, B, C, and D are not given, we cannot calculate the exact magnitude of dilation directly from that. However, based on the provided options and common knowledge:

   • If A', B', C', D' appear to be half the distance from the origin as A, B, C, D respectively, the magnitude of dilation would be \( \frac{1}{2} \).
   • If A', B', C', D' are one-third the distance from the origin compared to A, B, C, and D, the magnitude would be \( \frac{1}{3} \).
   • If A', B', C', D' are farther, such as doubling the distance, the magnitude would be 2.

   For the sake of this question, without more specific information or coordinates, let's assume the most common scenario in such questions where A'B'C'D' is smaller and the magnitude is \( \frac{1}{2} \).

2. Dilation of Triangle XYZ: The problem states that triangle XYZ with vertices: \( X(0, 4) \), \( Y(2, 0) \), and \( Z(4, 7) \) is dilated using a magnitude of \( \frac{1}{2} \) and center (4, 7).

   To calculate the new coordinates after dilation:

   • The formula for calculating the new coordinates after dilation is: \[ X' = (1 - k)(x - x_c) + x_c \] \[ Y' = (1 - k)(y - y_c) + y_c \] where \( k \) is the scale factor (in this case, \( \frac{1}{2} \)), \( (x_c, y_c) \) is the center of dilation \( (4, 7) \).

   Applying this to each vertex:

   • For \( X(0, 4) \): \[ X' = (1 - \frac{1}{2})(0 - 4) + 4 = \frac{1}{2}(-4) + 4 = -2 + 4 = 2 \] \[ Y' = (1 - \frac{1}{2})(4 - 7) + 7 = \frac{1}{2}(-3) + 7 = -1.5 + 7 = 5.5 \] So \( X' = (2, 5.5) \).

   • For \( Y(2, 0) \): \[ X' = (1 - \frac{1}{2})(2 - 4) + 4 = \frac{1}{2}(-2) + 4 = -1 + 4 = 3 \] \[ Y' = (1 - \frac{1}{2})(0 - 7) + 7 = \frac{1}{2}(-7) + 7 = -3.5 + 7 = 3.5 \] So \( Y' = (3, 3.5) \).

   • For \( Z(4, 7) \): It will remain the same as it is the center of dilation. So \( Z' = (4, 7) \).

   The new vertices after dilation are \( X' = (2, 5.5) \), \( Y' = (3, 3.5) \), \( Z' = (4, 7) \).

3. Dilation of Parallelogram ABCD with center (−2,−1) and magnitude of 2: To find the coordinates of point A after dilation with center at (-2, -1) and a magnitude of 2, we follow similar steps as before:

   • Let’s say the coordinates of point A are \( (x_a, y_a) \). The formula now reads: \[ x' = x_c + k(x - x_c) \quad \text{and} \quad y' = y_c + k(y - y_c) \] where \( k \) = 2. If we take one of the options and calculate, let's assume:
   • For \( A(-3, 0) \): \[ x' = -2 + 2(-3 + 2) = -2 + 2(-1) = -2 - 2 = -4 \] \[ y' = -1 + 2(0 + 1) = -1 + 2(1) = -1 + 2 = 1 \] So point A transforms to (-4, 1).

   After considering the math and ensuring it fits within the responses, the coordinates for point A after dilation would be matching the calculations above.

In summary:

1. The magnitude of dilation for the first question (parallelogram) is assumed to be \( \frac{1}{2} \).
2. Dilation of triangle XYZ results in \( X' = (2, 5.5) \), \( Y' = (3, 3.5) \), \( Z' = (4, 7) \).
3. The new location of point A after dilation will yield \( (−4, 1) \), leading to the choice being \( \left(−4, 3\right) \).

Please correct if any specific coordinates have been missed or if the values you began with differed from standard assumptions.