Question

write the coordinates of the vertices after a dilation with a scale factor of \\(\frac{1}{2}\\), centered at the origin.

Explanation:

Step1: Identify original coordinates

From the graph, the original vertices are: \( G(0, 10) \), \( F(2, -4) \), \( E(0, -10) \), \( H(-2, -4) \).

Step2: Apply dilation formula

The dilation formula centered at the origin with scale factor \( k \) is \( (x, y) \to (k \cdot x, k \cdot y) \), here \( k = \frac{1}{2} \).

• For \( G(0, 10) \):

\( x' = \frac{1}{2} \cdot 0 = 0 \), \( y' = \frac{1}{2} \cdot 10 = 5 \), so \( G'(0, 5) \).

• For \( F(2, -4) \):

\( x' = \frac{1}{2} \cdot 2 = 1 \), \( y' = \frac{1}{2} \cdot (-4) = -2 \), so \( F'(1, -2) \).

• For \( E(0, -10) \):

\( x' = \frac{1}{2} \cdot 0 = 0 \), \( y' = \frac{1}{2} \cdot (-10) = -5 \), so \( E'(0, -5) \).

• For \( H(-2, -4) \):

\( x' = \frac{1}{2} \cdot (-2) = -1 \), \( y' = \frac{1}{2} \cdot (-4) = -2 \), so \( H'(-1, -2) \).

Answer:

The coordinates after dilation are \( G'(0, 5) \), \( F'(1, -2) \), \( E'(0, -5) \), \( H'(-1, -2) \).