Question

The vertices of a rectangle are Q(−6,2),R(6,2),S(6,−4)Q(−6,2),R(6,2),S(6,−4), and T(−6,−4)T(−6,−4). Dilate the rectangle with respect to the origin using a scale factor of 3232. Then translate it 5 units right and 1 unit down. What are the coordinates of the image?

Answer 1

Multiply each x and y coordinate by the scale factor, so:
Q becomes: (-9, 3)
R becomes: (9, 3)
S becomes: (9, -6)
T becomes: (-9, -6)
Now, Add 5 to every x coordinate as we are moving 5 units in the positive x direction and take 1 from every y coordinate as we are moving 1 unit in the negative y direction, so:
Q becomes: (-4, 2)
R becomes: (14, 2)
S becomes: (14, -7)
T becomes: (-4, -7)

Answer 2

The coordinates of the image are Q''(-4,2), R''(14,2), S''(14,-7) and T''(-4,-7).

Explanation:

Given information: Q(−6,2),R(6,2),S(6,−4) and T(−6,−4).

If a figure dilated by factor k with respect to the origin , then

`(x,y)\rightarrow (kx,ky)`

The figure dilated with respect to the origin using a scale factor of 3/2.

`(x,y)\rightarrow ((3)/(2)x,(3)/(2)y)`

The vertices of rectangle after dilation are

`Q(-6,2)\rightarrow Q'(-9,3)`

`R(6,2)\rightarrow R'(9,3)`

`S(6,-4)\rightarrow S'(9,-6)`

`T(-6,-4)\rightarrow T'(-9,-6)`

Then translate it 5 units right and 1 unit down.

`(x,y)\rightarrow (x+5,y-1)`

`Q'(-9,3)\rightarrow Q''(-4,2)`

`R'(9,3)\rightarrow R''(14,2)`

`S'(9,-6)\rightarrow S''(14,-7)`

`T'(-9,-6)\rightarrow T''(-4,-7)`

Therefore the coordinates of the image are Q''(-4,2), R''(14,2), S''(14,-7) and T''(-4,-7).