Question

write the coordinates of the vertices after a dilation with a scale factor of \\(\frac{1}{3}\\), centered at the origin.\
\
(there is a coordinate grid with points b, c, d plotted. point b is at (0, 9), point c is at (9, 0)? wait, no, looking at the grid: the y-axis (vertical) has labels from -10 to 10, x-axis (horizontal) too. wait, the image shows a coordinate plane with grid lines, and points b, c, d. the text on the right says: write the coordinates of the vertices after a dilation with a scale factor of 1/3, centered at the origin. then there are placeholders for b, c, d coordinates: b( , ), c( , ), d( , ).)

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of points B, C, D. From the graph:

• Point B: \( (0, 9) \) (assuming each grid is 1 unit, y - coordinate is 9 as it's on y - axis, x = 0)
• Point C: \( (9, 0) \) (x - coordinate 9, y - coordinate 0 as it's on x - axis)
• Point D: \( (-9, 9) \) (x=-9, y = 9, since it's horizontally aligned with B and diagonally with C)

Step2: Apply dilation formula

The rule for dilation with scale factor \( k=\frac{1}{3} \) centered at the origin is \( (x,y)\to(kx,ky) \)

For point B:

\( x = 0,y = 9 \)
New coordinates \( B'=( \frac{1}{3}\times0,\frac{1}{3}\times9)=(0, 3) \)

For point C:

\( x = 9,y = 0 \)
New coordinates \( C'=( \frac{1}{3}\times9,\frac{1}{3}\times0)=(3, 0) \)

For point D:

\( x=-9,y = 9 \)
New coordinates \( D'=( \frac{1}{3}\times(-9),\frac{1}{3}\times9)=(-3, 3) \)

Answer:

• \( B'=(0, 3) \)
• \( C'=(3, 0) \)
• \( D'=(-3, 3) \)