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:

• \( T(-8, 2) \)
• \( U(4, 2) \)
• \( V(4, -10) \)
• \( S(-8, -10) \)

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 \( T(-8, 2) \):
\( x' = \frac{1}{2} \times (-8) = -4 \)
\( y' = \frac{1}{2} \times 2 = 1 \)
So, \( T'(-4, 1) \)

For \( U(4, 2) \):
\( x' = \frac{1}{2} \times 4 = 2 \)
\( y' = \frac{1}{2} \times 2 = 1 \)
So, \( U'(2, 1) \)

For \( V(4, -10) \):
\( x' = \frac{1}{2} \times 4 = 2 \)
\( y' = \frac{1}{2} \times (-10) = -5 \)
So, \( V'(2, -5) \)

For \( S(-8, -10) \):
\( x' = \frac{1}{2} \times (-8) = -4 \)
\( y' = \frac{1}{2} \times (-10) = -5 \)
So, \( S'(-4, -5) \)

Answer:

The coordinates of the vertices after dilation are \( T'(-4, 1) \), \( U'(2, 1) \), \( V'(2, -5) \), and \( S'(-4, -5) \).