Question

Use the image to answer the question.

Smaller parallelogram upper A prime upper B prime upper C prime upper D prime is graphed within larger parallelogram upper A upper B upper C upper D on a coordinate plane.

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

13
Start Fraction 1 over 3 End Fraction

1
1

12
Start Fraction 1 over 2 End Fraction

2
2
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Coordinates are A is (-2,4) B is (4,4) A'(-1,2) B'(2,2) D' is (-2,-2) D is (-4,-2) C' is (1,-1) C is (2,-2)

Answer 1

To find the magnitude of dilation from parallelogram ABCD to parallelogram A'B'C'D', we can use a point from both the original and dilated shapes.

Using the coordinates provided:
- Point A: (-2, 4)
- Point A': (-1, 2)

The formula for dilation in reference to the origin is given by the ratio of the distances of the points from the origin.

1. Calculate the distance of point A from the origin:
\[
d(A) = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

2. Calculate the distance of point A' from the origin:
\[
d(A') = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\]

3. Now, calculate the magnitude of dilation (k):
\[
k = \frac{d(A')}{d(A)} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}
\]

Therefore, the magnitude of dilation is \( \frac{1}{2} \).

The correct response is:
**Start Fraction 1 over 2 End Fraction**.