Question

Find the images of the vertices of PQR shown below, for a dilation with center (0,0) and scale factor of 2. Then graph the image.

Answer 1

The images of the vertices of PQR are P' (8, -6), Q' (8, 6), and R' (14, -6).

A graph that shows triangle PQR and the image triangle P'Q'R is shown in the picture below.

In Mathematics and Geometry, a dilation is a type of transformation which typically changes the size (dimensions) of a geometric object, but not its shape.

In this scenario an exercise, we would dilate the coordinates of the pre-image (triangle PQR) by applying a scale factor of 2 that is centered at the origin (0, 0) in order to produce the coordinates of its image as follows:

Vertices P (4, -3) → (4 × 2, -3 × 2) = P' (8, -6).

Vertices Q (4, 3) → (4 × 2, 3 × 2) = Q' (8, 6).

Vertices R (7, -3) → (7 × 2, -3 × 2) = R' (14, -6).

In concluson, we can logically deduce that the dilation is an enlargement because the size of the pre-image is smaller than the image.

Answer 2

The images of PQR are P'=(8 , -6) , Q' = (8,6) and R' = (14 , -6)

Explanation:

From the given graph:

the coordinates of PQR i.e,

P= (4, -3) , Q= (4,3) , R= (7,-3)

Given: the scale factor of 2 and center of dilation at origin i.e (0,0)

The mapping rule for the dilation applied to the triangle as shown below:

`(x, y) \rightarrow ( kx, ky)` where k represents the scale factor i.e, k=2 or we can write it as ;

`(x, y) \rightarrow (2x, 2y)`

For P = (4, -3)

The image P' =
`(2\cdot 4 , 2\cdot (-3))`

⇒ P'=
`(8, -6)`

Similarly for Q = (4, 3) and R = (7 , -3)

the image of Q' =
`(2\cdot 4 , 2\cdot (3)) = (8 , 6)`

The image of R' =
`(2\cdot 7 , 2\cdot (-3)) = (14 , -6)`

Now, plot the graph of PQR as shown in the attachment: