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

1
1

2
2

12
Start Fraction 1 over 2 End Fraction

13
Start Fraction 1 over 3 End Fraction
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Answer 1

To determine the magnitude of dilation from parallelogram ABCD to parallelogram A′B′C′D′, we need to compare the corresponding vertices of both parallelograms.

If the coordinates of a point in parallelogram ABCD are (x, y) and the corresponding point in A′B′C′D′ is (x', y'), the formula for dilation can be expressed as:

\[
x' = k \cdot x
\]
\[
y' = k \cdot y
\]

where \( k \) is the scale factor of dilation.

To find \( k \), we can examine the coordinates of points from both parallelograms. If, for example, we had a point A at (x, y) and point A' at (x', y'), we would solve for \( k \) using the ratio of the coordinates:

\[
k = \frac{x'}{x} = \frac{y'}{y}
\]

Without the specific coordinates provided in the image, I cannot calculate the exact value of \( k \). However, you can apply this reasoning to the coordinates of the points of A, B, C, and D to calculate the appropriate dilation factor.

If the dilation results in the smaller parallelogram being half the size of the original, the factor \( k \) would be \( \frac{1}{2} \), which corresponds to the option "12 Start Fraction 1 over 2 End Fraction".

If the ratio shows that the smaller parallelogram is one-third the size of the larger one, then the factor would be \( \frac{1}{3} \), corresponding to the option “13 Start Fraction 1 over 3 End Fraction”.

Please check the vertices you'll be using for the calculation, and see if you can apply the ratio formula to determine the dilation factor.