Question

write the coordinates of the vertices after a dilation with a scale factor of \\(\frac{1}{4}\\), centered at the origin.

Explanation:

Step1: Identify original coordinates

From the graph, the original coordinates are:

• \( P(4, -8) \)
• \( Q(4, 4) \)
• \( R(8, 8) \)
• \( S(8, -4) \)

Step2: Apply dilation formula

Dilation centered at the origin with scale factor \( k \) transforms a point \( (x, y) \) to \( (k \cdot x, k \cdot y) \). Here, \( k = \frac{1}{4} \).

For \( P(4, -8) \):
\( x' = \frac{1}{4} \cdot 4 = 1 \), \( y' = \frac{1}{4} \cdot (-8) = -2 \) → \( P'(1, -2) \)

For \( Q(4, 4) \):
\( x' = \frac{1}{4} \cdot 4 = 1 \), \( y' = \frac{1}{4} \cdot 4 = 1 \) → \( Q'(1, 1) \)

For \( R(8, 8) \):
\( x' = \frac{1}{4} \cdot 8 = 2 \), \( y' = \frac{1}{4} \cdot 8 = 2 \) → \( R'(2, 2) \)

For \( S(8, -4) \):
\( x' = \frac{1}{4} \cdot 8 = 2 \), \( y' = \frac{1}{4} \cdot (-4) = -1 \) → \( S'(2, -1) \)

Answer:

\( P'(1, -2) \), \( Q'(1, 1) \), \( R'(2, 2) \), \( S'(2, -1) \)