Question

the figure below is dilated by a factor of 4 centered at the origin. plot the resulting image. click twice to plot a segment. click a segment to delete it.

Explanation:

Step1: Identify original coordinates

First, we need to find the coordinates of the original vertices. Let's assume the original vertices are:

• \( Q \): Let's say \( Q = (-2, 0) \) (from the grid, assuming each grid square is 1 unit)
• \( P \): \( P = (-1, 1) \)
• \( T \): \( T = (1, 1) \)
• \( S \): \( S = (2, 0) \)
• \( R \): \( R = (0, -2) \) (Wait, maybe I misread. Let's re - examine. Wait, the figure seems to be a polygon with vertices. Let's correctly identify the coordinates. Let's take the origin as (0,0). Let's look at the x - axis (horizontal) and y - axis (vertical). Let's assume the original vertices:

Let's say \( Q = (-2, 0) \), \( P = (-1, 1) \), \( T=(1, 1) \), \( S=(2, 0) \), \( R=(0, - 2) \) (maybe it's a pentagon or a quadrilateral? Wait, the figure is a polygon, let's list the coordinates properly.

Step2: Apply dilation rule

The rule for dilation centered at the origin with a scale factor \( k \) is \( (x,y)\to(kx,ky) \). Here, \( k = 4 \).

• For point \( Q(-2,0) \):

New \( x \) - coordinate: \( 4\times(-2)=-8 \)
New \( y \) - coordinate: \( 4\times0 = 0 \)
So, \( Q'=(-8,0) \)

• For point \( P(-1,1) \):

New \( x \) - coordinate: \( 4\times(-1)=-4 \)
New \( y \) - coordinate: \( 4\times1 = 4 \)
So, \( P'=(-4,4) \)

• For point \( T(1,1) \):

New \( x \) - coordinate: \( 4\times1 = 4 \)
New \( y \) - coordinate: \( 4\times1 = 4 \)
So, \( T'=(4,4) \)

• For point \( S(2,0) \):

New \( x \) - coordinate: \( 4\times2 = 8 \)
New \( y \) - coordinate: \( 4\times0 = 0 \)
So, \( S'=(8,0) \)

• For point \( R(0,-2) \):

New \( x \) - coordinate: \( 4\times0 = 0 \)
New \( y \) - coordinate: \( 4\times(-2)=-8 \)
So, \( R'=(0, - 8) \)

Step3: Plot the new points

Now, we plot the new points \( Q'(-8,0) \), \( P'(-4,4) \), \( T'(4,4) \), \( S'(8,0) \), \( R'(0, - 8) \) on the coordinate plane. Connect the points in the same order as the original figure to get the dilated image.

Answer:

To plot the dilated image, we first find the coordinates of the original vertices, then apply the dilation rule \( (x,y)\to(4x,4y) \) to each vertex, and then plot the new vertices \( Q'(-8,0) \), \( P'(-4,4) \), \( T'(4,4) \), \( S'(8,0) \), \( R'(0, - 8) \) (or other vertices depending on the exact original figure) and connect them.