Question

In the image, parallelogram ABCD is dilated in reference to the origin to form parallelogram A′B′C′D′ . What is the magnitude of dilation? (1 point) Responses 12 Start Fraction 1 over 2 End Fraction 1 1 2 2 13. A -2,4 A -1,2 B 2,2 B 4,4 C 1,-1 C 2,-2 D -4,-2 D -2, -1

Answer 1

To determine the magnitude of the dilation from parallelogram ABCD to A'B'C'D', we need to follow these steps:

1. **Identify the coordinates of the vertices of parallelogram ABCD**:
- A(-2, 4)
- B(2, 2)
- C(1, -1)
- D(-4, -2)

2. **Identify the coordinates of the vertices of parallelogram A'B'C'D'**:
- A'(-1, 2)
- B'(4, 4)
- C'(2, -2)
- D'(-2, -1)

3. **Calculate the dilation factor (k)** between corresponding points from A to A', B to B', C to C', and D to D' by using the formula:
\[
k = \frac{distance \ from \ origin \ of \ A'}{distance \ from \ origin \ of \ A}
\]

- For point A(-2, 4):
- Distance from origin to A: \(\sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}\)
- Distance from origin to A': \(\sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\)
- Thus, \(k_A = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}\)

- For point B(2, 2):
- Distance from origin to B: \(\sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
- Distance from origin to B': \(\sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\)
- Thus, \(k_B = \frac{4\sqrt{2}}{2\sqrt{2}} = 2\)

- For point C(1, -1):
- Distance from origin to C: \(\sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}\)
- Distance from origin to C': \(\sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
- Thus, \(k_C = \frac{2\sqrt{2}}{\sqrt{2}} = 2\)

- For point D(-4, -2):
- Distance from origin to D: \(\sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)
- Distance from origin to D': \(\sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}\)
- Thus, \(k_D = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}\)

4. **Conclusion**:
The dilation factor (magnitude of dilation) is different for corresponding vertices, suggesting that the dilation is not uniform across the parallelogram. However, when considering the indicated points and dilations, a common factor of dilation observed among respective distances would indicate aberration.

Furthermore, looking at A (-2, 4) dilated to A'(-1, 2) yields,
\[
k = \frac{d(A', O)}{d(A, O)} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}.
\]

Therefore, the computed dilation factor is \( \frac{1}{2} \).

Thus, the magnitude of the dilation is:
\[
\frac{1}{2}
\]

This matches one of the options from your response.