Question

List the sequence of transformations that must occur to transform rectangle ABCD into A'B'C'D

Answer 1

Rectangle ABCD was dilated by
`(1)/(2)` followed by a translation of 7 units right and 1 unit down.

Explanation:

Rectangle A'B'C'D' is a dilated form of rectangle ABCD.

Followed by a translation, as shown in the figure attached.

For dilation,

Scale factor =
`\frac{\text{Side length of image}}{\text{Side length of original}}`

=
`(2)/(4)`

=
`(1)/(2)`

Therefore, rectangle ABCD was dilated with a scale factor of
`(1)/(2)` about the origin.

Coordinates of point A → (-10, 6)

Coordinates of point A' → (1, 2)

Coordinates of image of point A after dilation,

(x, y) → (kx, ky)

A(-10, 6) → A"(-5, 3)

Coordinates of point A" after translation of 'h' units to the right and 'k' units down.

(x, y) → (x + h, y - k)

A"(-5, 3) → A'[(-5 + h), (3 - k)]

By comparing the coordinates of A'[(1, 2) and (-5 + h, 3 - k)],

-5 + h = 2 ⇒ h = 7

3 - k = 2 ⇒ k = 1

Therefore, rectangle ABCD was dilated by
`(1)/(2)` followed by a translation of 7 units right and 1 unit down.