Question

Lyndsey’s advertising firm is working on a project for one of its customers. Lyndsey is asked to move a figure in the advertisement. The new figure must be similar to the old figure. The change is shown in the graph. Write a series of transformations that Lyndsey can use to move the old figure to the new figure. Explain why the figures are similar. *

Answer 1

Explanation:

Since, old figure is the dilated form of the new figure.

For the dilation of the old figure to new by a scale factor 's',

s =
`\frac{\text{Side length of the new figure}}{\text{Side length of the old figure}}` =
`(2)/(4)=(1)/(2)`

Rule for the dilation of a point by a scale factor 'k',

(x, y) → k(x, y)

→ (kx, ky)

If the vertices of the old triangle are dilated by a scale factor of
`(1)/(2)`,

Coordinates of the dilated triangle of the old triangle will be,

A(3, 4) → A'(
`(3)/(2)`, 2)

B(3, 8) → B'(
`(3)/(2)`, 4)

C(5, 4) → C'(
`(5)/(2)`, 2)

Now these points are rotated by an angle of 90° counterclockwise about the origin,

Rule for the rotation is,

(x, y) → (-y, x)

`A'((3)/(2),2)` → A"(-2,
`(3)/(2)`)

B'(
`(3)/(2)`, 4) → B"(-4,

C'(
`(5)/(2)`, 2) → C"(-2,

Further these points have been reflected across x-axis,

A"(-2,
`(3)/(2)`) → P(-2, -
`(3)/(2)`)

B"(-4,
`(3)/(2)`) → Q(-4, -
`(3)/(2)`)

C"(-2,
`(5)/(2)`) → R(-2, -
`(5)/(2)`)

Therefore, old triangle is dilated by a scale factor of
`(1)/(2)`, followed by rotation of 90° counterclockwise about the origin and then reflected across x-axis to form the new triangle.