Question

write the coordinates of the vertices after a dilation with a scale factor of \\(\frac{1}{3}\\), centered at the origin.
\\(b(\square, \square)\\)
\\(c(\square, \square)\\)
\\(d(\square, \square)\\)

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of points \( B \), \( C \), and \( D \) from the graph.

• For point \( B \): Looking at the graph, \( B \) is at \( (-3, -9) \)? Wait, no, let's check the grid. Wait, the x - coordinate: each grid is 1 unit. Let's see, \( B \) is at \( x=-3 \)? Wait, no, looking at the graph, \( B \) is at \( (-3, -9) \)? Wait, no, let's re - examine. Wait, the vertical line for \( B \) and \( C \): \( B \) is at \( (-3, -9) \)? Wait, no, the grid lines: from the origin, moving left 3 units (x = - 3) and down 9 units (y=-9)? Wait, no, the y - axis: the bottom is - 10, so \( B \) is at \( (-3, -9) \)? Wait, no, looking at the graph, \( B \) is at \( (-3, -9) \)? Wait, no, let's check the coordinates again. Wait, the point \( B \): x - coordinate is - 3, y - coordinate is - 9? Wait, no, the horizontal line for \( B \) and \( D \): \( B \) is at \( (-3, -9) \), \( D \) is at \( (9, -9) \), and \( C \) is at \( (-3, 6) \).

Step2: Apply dilation with scale factor \( \frac{1}{3} \) centered at the origin

The rule for dilation centered at the origin with scale factor \( k \) is \( (x,y)\to(kx,ky) \).

For point \( B(-3,-9) \):

Multiply each coordinate by \( \frac{1}{3} \).
\( x'=\frac{1}{3}\times(-3)= - 1 \)
\( y'=\frac{1}{3}\times(-9)=-3 \)
So \( B'(-1, - 3) \)

For point \( C(-3,6) \):

\( x'=\frac{1}{3}\times(-3)=-1 \)
\( y'=\frac{1}{3}\times6 = 2 \)
So \( C'(-1,2) \)

For point \( D(9,-9) \):

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

Answer:

\( B'(-1, - 3) \), \( C'(-1, 2) \), \( D'(3, - 3) \)