Answer key: Transformations Worksheet: Translate, Rotate & Reflect

For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: Transformations Worksheet: Translate, Rotate & Reflect.

Goal: Apply translations, rotations and reflections to a triangle on the coordinate plane, write the coordinate rule for each, and see which transformations keep the size the same.

Connects to Common Core 8.G.A.3 (Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.); HSG-CO.A.5 (Given a geometric figure and a rotation, reflection or translation, draw the transformed figure.)

Completed results table

Values are the readouts of the tool (Transformation playground) for each setting.

Answers

1. Write the coordinate rule for the translation you used. Did the area change?

   (x, y) → (x + 2, y + 1). For example, A(−2, −1) moves to A′(0, 0). The area stays 6 square units: a translation slides the triangle without changing its size or shape.

2. Look at the 90° rotation. Find the rule that turns each original point into its image.

   (x, y) → (−y, x) for a 90° counterclockwise rotation about the origin. For example, A(−2, −1) → A′(1, −2) and C(0, 2) → C′(−2, 0).

3. Compare the 180° rotation with two reflections in a row: first over the x-axis, then over the y-axis. What do you find?

   They give the same image: A′(2, 1), B′(−2, 1), C′(0, −2). Reflecting over the x-axis sends (x, y) to (x, −y); reflecting that over the y-axis gives (−x, −y), which is the rule for a 180° rotation.

4. Which transformations kept the area at 6 square units? Which one changed it, and by what factor?

   Translations, rotations and reflections are rigid motions: they keep every length, every angle and the area (6 square units). The dilation with scale factor 2 changed the area to 24 square units, 4 times as large, because area grows by the square of the scale factor (2² = 4).

5. Predict where C(0, 2) lands after a 270° counterclockwise rotation. Then set the rotation angle to 270 to check.

   A 270° rotation follows the rule (x, y) → (y, −x), so C(0, 2) → C′(2, 0). The tool agrees.

6. In the original triangle, going A → B → C runs counterclockwise. Which transformations in your table reverse this to clockwise?

   Only the two reflections. They flip the triangle over, so A′ → B′ → C′ runs clockwise. Translations, rotations and the dilation keep the counterclockwise order.