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

A(-2,4) to A'(-1,2)
B(4,4) to B'(2,2)
C(2,-2) to C'(1,-1)
D(-4,-2) to D'(-2,-1)

Answer 1

To find the magnitude of dilation from the original parallelogram ABCD to the dilated parallelogram A'B'C'D', we can use the formula for dilation. If a point \( P(x, y) \) is dilated to a point \( P'(x', y') \) in reference to the origin with a scale factor \( k \), then:

\[
P' = k \cdot P
\]

For each point, we can calculate the scale factor \( k \) by taking the ratio of the coordinates of the dilated point to the original point.

### Let's calculate the scale factor for each point.

1. **For point A (-2, 4) to A' (-1, 2):**
\[
k_A = \frac{x'}{x} = \frac{-1}{-2} = \frac{1}{2}
\]
\[
k_A = \frac{y'}{y} = \frac{2}{4} = \frac{1}{2}
\]
Both x and y give the same scale factor \( k_A = \frac{1}{2} \).

2. **For point B (4, 4) to B' (2, 2):**
\[
k_B = \frac{x'}{x} = \frac{2}{4} = \frac{1}{2}
\]
\[
k_B = \frac{y'}{y} = \frac{2}{4} = \frac{1}{2}
\]
Both x and y give the same scale factor \( k_B = \frac{1}{2} \).

3. **For point C (2, -2) to C' (1, -1):**
\[
k_C = \frac{x'}{x} = \frac{1}{2} = \frac{1}{2}
\]
\[
k_C = \frac{y'}{y} = \frac{-1}{-2} = \frac{1}{2}
\]
Both x and y give the same scale factor \( k_C = \frac{1}{2} \).

4. **For point D (-4, -2) to D' (-2, -1):**
\[
k_D = \frac{x'}{x} = \frac{-2}{-4} = \frac{1}{2}
\]
\[
k_D = \frac{y'}{y} = \frac{-1}{-2} = \frac{1}{2}
\]
Both x and y give the same scale factor \( k_D = \frac{1}{2} \).

### Conclusion
In all cases, the scale factor \( k \) is consistent:

\[
k = \frac{1}{2}
\]

Thus, the magnitude of dilation for parallelogram ABCD to parallelogram A'B'C'D' is \( \frac{1}{2} \).