Question

1. parallelogram w is graphed on a coordinate plane and is dilated by a scale factor of \\(\frac{2}{3}\\) about the origin to form parallelogram z. which statement is true?

a parallelogram z is larger than parallelogram w because scale factors between 0 and 1 create larger shapes since the distances between the vertices in the image figure are increased.
b parallelogram z is larger than parallelogram w because scale factors between 0 and 1 create larger shapes since the distances between the vertices in the image figure are decreased.
c parallelogram z is smaller than parallelogram w because scale factors between 0 and 1 create smaller shapes since the distances between the vertices in the image figure are decreased.
d parallelogram z is smaller than parallelogram w because scale factors between 0 and 1 create smaller shapes since the distances between the vertices in the image figure are increased.
\\(\bigcirc\\) a
\\(\bigcirc\\) b
\\(\bigcirc\\) c
\\(\bigcirc\\) d

Explanation:

Step1: Recall Dilation Scale Factor Effect

A dilation with a scale factor \( k \) about the origin multiplies the coordinates of each vertex by \( k \). If \( 0 < k < 1 \), the distances between vertices (and thus the size of the figure) decrease, making the image smaller than the original. If \( k > 1 \), the distances increase, making the image larger. Here, the scale factor is \( \frac{2}{3} \), which is between 0 and 1.

Step2: Analyze Each Option

• Option A: Claims scale factors between 0 and 1 create larger shapes. This is false, as \( 0 < k < 1 \) reduces size.
• Option B: Claims scale factors between 0 and 1 create larger shapes (false) and says distances increase (false, they decrease).
• Option C: States parallelogram Z is smaller (correct, since \( \frac{2}{3} < 1 \)) and explains scale factors between 0 and 1 create smaller shapes because distances between vertices are decreased (correct, as multiplying by \( \frac{2}{3} \) reduces distances).
• Option D: Claims distances between vertices are increased (false, they are decreased).

Answer:

C. Parallelogram Z is smaller than parallelogram W because scale factors between 0 and 1 create smaller shapes since the distances between the vertices in the image figure are decreased.