Question

What set of transformations could be applied to rectangle ABCD to create A’B’C’D’?

A)Reflected over the x-axis and rotated 180 degrees

B)Reflected over the y-axis and rotated 180 degrees

C)Reflected over the x-axis and rotated 90 degrees counterclockwise

D)Reflected over the y-axis and rotated 90 degrees counterclockwise

Answer 1

Rectangle ABCD has vertices A(-4,2), B(-4,1), C(-1,1) and D(-1,2).

1. Apply the reflection over the x-axis that has a rule

(x,y)→(x,-y).

Then

• A(-4,2)→A''(-4,-2);
• B(-4,1)→B''(-4,-1);
• C(-1,1)→C''(-1,-1);
• D(-1,2)→D''(-1,-2).

2. The second transformation is rotation by 90° counterclockwise about the origin. This transformation has a rule

(x,y)→(-y,x).

Then

• A''(-4,-2)→A'(2,-4);
• B''(-4,-1)→B'(1,-4);
• C''(-1,-1)→C'(1,-1);
• D''(-1,-2)→D'(2,-1).

As you can see these points are exactly those frome the diagram.

Answer: 1st transformation is reflection over the x-axis and second transformation is rotation by 90° counterclockwise about the origin.