We are told that trapezoid ABCD is dilated using a scale factor of 3, and then it is rotated 180^(∘) about the origin. We want to find the coordinates of the final image. To do so, we will do one transformation at time!
Dilation
A dilation can be an enlargement or a reduction of the preimage. Which type of dilation it is depends on the value of the scale factor k.
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Preimage & & Image [0.5em]
(x,y)& ⇒ & ( kx, ky)
Now let's find the coordinates of the vertices of ABCD after a dilation with a scale factor k= 3.
Dilation With Scale Factor k=3
Preimage
Multiply by k
Image
A(- 2, - 1)
( 3(- 2), 3(- 1))
A'(- 6,- 3)
B(- 1 ,1)
( 3(- 1), 3(1))
B'(- 3,3)
C(0, 1)
( 3(0), 3(1))
C'(0,3)
D(0, - 1)
( 3(0), 3(- 1))
D'(0,- 3)
We can now plot the new points and connect them with segments to draw the image.
Rotation
A rotation is a transformation about a fixed point called center of rotation. Each point of the original figure and its image are the same distance from the center of rotation. When a clockwise rotation is performed about the origin, the coordinates of the image can be written in relation to the coordinates of the preimage.
We want to rotate trapezoid A'B'C'D' 180^(∘) clockwise about the origin. Let's use the information in the table to find the coordinates of the image of each vertex.
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Preimage & & Image
(x,y) & → & (- x,- y) [0.5em]
A'(- 6,- 3) & & A''(6,3) [0.5em]
B'(- 3, 3) & & B''(3,- 3) [0.5em]
C'(0,3) & & C''(0,- 3) [0.5em]
D'(0,- 3) & & D''(0,3)
We can now plot the final points and draw the image after the rotation!
