Exercise 32 Page 699

Practice makes perfect

a To draw the image of △ DEF under a dilation with center D and scale factor 3, we will follow three steps.

Draw a ray from D through the vertex E and locate E' on DE such that DE'= 3 DE.
Locate the other points in the same way.
Draw △ D'E'F'.

Note that D is the center of the dilation. As a result, its image is itself. Now let's draw the graph of the dilated image.

b Let's first draw the image of △ DEF under a dilation with a scale factor of 3 centered at the origin. To do so, we multiply the x- and y-coordinates of each vertex by the scale factor.

lcl (x,y) & → & (3x, 3y) [0.6em] [-0.6em] D(1,2) & → & D'(3,6) [0.5em] E(3,2) & → & E'(9,6) [0.5em] F(3,0) & → & F'(9,0)

Let's plot the points and draw the image of △ DEF.

Next, we compare this image with the graph of the image in Part A.

These images coincide if △ D'E'F' is translated 4 units down and 2 units left. Therefore, a dilation with a scale factor of 3 centered at the origin followed by a translation along ⟨ - 2,- 4 ⟩ describes the dilation in Part A.

c In Parts B we have shown that a dilation by a scale factor of 3 with a center of dilation D(1,2) is equivalent with a dilation centered at the origin followed by a translation along ⟨ - 2, - 4 ⟩. We can also see this in the applet below.

Notice that we can rewrite the translation vector using the coordinates of D(1,2). ⟨ - 2, - 4 ⟩ ⇕ ⟨ - 2( 1), - 2( 2) ⟩ This is true for other points, too. Therefore, if a figure is dilated by a scale factor of 3 with a center D(x,y), the same image can be created through a dilation with center at the origin and scale factor 3 followed by a translation along the vector ⟨ - 2 x, - 2 y ⟩.

Subchapter links

Exercises p.697-701

Exercise 32 Page 699

Hints

Check the answer

Practice exercises

Asses your skill