Chanel says the sequence of transformations that maps trapezoid PQRS onto trapezoid KLMN is: rotate it 90 degrees counterclockwise, and then dilate it by a scale factor of 1/2. Preston says the sequence of transformations that maps trapezoid PQRS onto trapezoid KLMN is: reflect it across the x-axis, and then dilate it by a scale factor of 1/2.

Part A

Start with Chanel’s sequence. How does a 90-degree counterclockwise rotation about the origin change the coordinates of a shape?

Part B

Complete the table to find the rule for the rotation, the coordinates of trapezoid PQRS, and the coordinates of the rotated image, trapezoid P′Q′R′S′.
Original Coordinates New Coordinates
(x, y) ( , )
P( , ) P′( , )
Q( , ) Q′( , )
R( , ) R( , )
S( , ) S′( , )

Part C

How does a dilation by a scale factor of change the coordinates of a shape?

Part D

Complete the table to find the rule for the dilation, the coordinates of trapezoid P′Q′R′S′, and the coordinates of the dilated image, trapezoid P″Q″R″S″.
Original Coordinates New Coordinates
(x, y) ( , )
P′( , ) P″( , )
Q′( , ) Q″( , )
R′( , ) R″( , )
S′( , ) S″( , )

Part E

Now look at Preston’s sequence. How does a reflection across the x-axis change the coordinates of a shape?

Part F

Complete the table to find the rule for the reflection, the coordinates of trapezoid PQRS, and the coordinates of the reflected image, trapezoid P′Q′R′S′.
Original Coordinates New Coordinates
(x, y) ( , )
P( , ) P′( , )
Q( , ) Q′( , )
R( , ) R′( , )
S( , ) S′( , )

Part G

How does a dilation by a scale factor of change the coordinates of a shape?

Part H

Complete the table to find the rule for the dilation, the coordinates of trapezoid P′Q′R′S′, and the coordinates of the dilated image, trapezoid P″Q″R″S″.
Original Coordinates New Coordinates
(x, y) ( , )
P′( , ) P″( , )
Q′( , ) Q″( , )
R′( , ) R″( , )
S′( , ) S″( , )

Part I

Whose sequence proves trapezoid PQRS is similar to trapezoid KLMN? Why?