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

Answer 1

To find the magnitude of the dilation from parallelogram ABCD to A'B'C'D', we can analyze the relationship between the corresponding points.

Dilation is defined as the transformation of points in a plane relative to a center point (in this case, the origin) by a scale factor. Mathematically, for a point \( (x, y) \) being dilated to a point \( (x', y') \) in reference to the origin by a scale factor \( k \), this relationship can be expressed as:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

We can find the scale factor \( k \) by using the coordinates of corresponding points.

Let's take point A and A':
- Original point A: \( (-2, 4) \)
- Dilated point A': \( (-1, 2) \)

Using the formula:

\[
(-1, 2) = (k \cdot -2, k \cdot 4)
\]

So we have the equations:
1. \( k \cdot -2 = -1 \)
2. \( k \cdot 4 = 2 \)

From the first equation, solving for \( k \):
\[
k \cdot -2 = -1 \quad \Rightarrow \quad k = \frac{-1}{-2} = \frac{1}{2}
\]

From the second equation, solving for \( k \):
\[
k \cdot 4 = 2 \quad \Rightarrow \quad k = \frac{2}{4} = \frac{1}{2}
\]

In both cases, we find that \( k = \frac{1}{2} \).

This indicates that the magnitude of dilation from parallelogram ABCD to A'B'C'D' is:

\[
\boxed{\frac{1}{2}}
\]