Transformations

Grade 8 · geometry · 66 practice problems · read aloud

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Transformations: Moving Shapes

Transformations change a shape's position or size on a coordinate plane. They are useful in design, animation, and understanding symmetry. The three main types are translations, reflections, and rotations.

Step-by-Step Guide

  1. Identify the Type: Is it a slide (translation), flip (reflection), or turn (rotation)?
  2. Apply the Rule:
    • Translation: Add to (slide right/up) or subtract from (slide left/down) the x and y coordinates. Example: (x, y) → (x + 5, y - 2)
    • Reflection: Multiply the coordinate you are reflecting over by -1.
      • Over x-axis: (x, y) → (x, -y)
      • Over y-axis: (x, y) → (-x, y)
    • Rotation: Turn the shape around a point (usually the origin).
      • 90° clockwise: (x, y) → (y, -x)
      • 180°: (x, y) → (-x, -y)
  3. Plot the New Points: Calculate the new coordinates for each vertex and connect them.

Visual Examples

Example 1: Translation

Translate triangle ABC with vertices A(1,2), B(3,2), C(2,4) by the rule (x, y) → (x + 4, y + 1).

Step 1: Apply the rule to each point: A(1,2) → (1+4, 2+1) = A'(5,3) B(3,2) → (3+4, 2+1) = B'(7,3) C(2,4) → (2+4, 4+1) = C'(6,5)

Step 2: Plot points A', B', C' and connect them. The shape slides 4 right and 1 up.

Example 2: Reflection

Reflect point D(2, -3) over the x-axis.

Step 1: Use the rule for x-axis reflection: (x, y) → (x, -y).

Step 2: Apply the rule: D(2, -3) → D'(2, -(-3)) = D'(2, 3). The point flips vertically.

Common Mistakes ⚠️

Mixing up x and y changes: For translations, remember (x, y). Moving right/left changes x. Moving up/down changes y.

Reflection confusion: Reflecting over the x-axis changes the y-sign. Reflecting over the y-axis changes the x-sign.

Rotation direction: 90° clockwise is different from 90° counterclockwise. Always note the direction.

Tips & Tricks

Translation: Think "slide." The shape doesn't change size or orientation.

Reflection: Think "mirror" or "flip." The shape is reversed.

Rotation: Think "spin." The shape turns around a point.

Memory Aid: For 90° clockwise rotation, just swap the coordinates and make the new y negative: (x, y) → (y, -x).

Practice Suggestions

  • Start by transforming single points before moving to whole shapes.
  • Use graph paper or digital tools to draw the original and transformed shapes.
  • Create your own shapes on a coordinate grid and practice all three transformations.
  • Check your work by seeing if the size and shape (the congruence) is preserved.

Practice problems

6 of the 66, worked through step by step — try them before opening the answer.

1 Triangle ABC with vertices A(2, 7), B(5, 7), C(2, 3) is reflected over the y-axis. What are the coordinates of A'?

Hint: When reflecting over the y-axis, the x-coordinate changes sign while the y-coordinate stays the same.

Show the answer

Answer: (-2, 7)

  1. The original coordinates of point A are (2, 7).
  2. When reflecting over the y-axis, the x-coordinate changes sign.
  3. The new x-coordinate becomes -2.
  4. The y-coordinate remains the same: 7.
  5. Therefore, the coordinates of A' are (-2, 7).

2 Triangle ABC with vertices A(2,4), B(6,4), C(4,8) is reflected over the y-axis. What are the coordinates of A'B'C'?

Hint: When reflecting over the y-axis, the x-coordinates change sign while the y-coordinates stay the same.

Show the answer

Answer: (-2,4), (-6,4), (-4,8)

  1. For reflection over the y-axis, the rule is (x,y) → (-x,y)
  2. Apply this to point A(2,4): A' = (-2,4)
  3. Apply this to point B(6,4): B' = (-6,4)
  4. Apply this to point C(4,8): C' = (-4,8)
  5. The coordinates of A'B'C' are (-2,4), (-6,4), (-4,8)

3 Triangle ABC with vertices A(3,5), B(7,5), C(5,9) is reflected over the y-axis. What are the coordinates of A'B'C'?

Hint: When reflecting over the y-axis, the x-coordinates change sign while the y-coordinates remain the same.

Show the answer

Answer: A'(-3,5), B'(-7,5), C'(-5,9)

A reflection over the y-axis is a transformation that flips a figure across the vertical axis. This transformation preserves the shape and size of the figure, making it congruent to the original. The rule for reflecting over the y-axis is (x,y) → (-x,y).

4 Triangle ABC with vertices A(2,7), B(7,2), C(12,7) is reflected over the x-axis. What are the coordinates of A'B'C'?

Hint: When reflecting over the x-axis, the x-coordinates stay the same while the y-coordinates change sign.

Show the answer

Answer: A(2,-7), B(7,-2), C(12,-7)

A reflection over the x-axis is a transformation that flips a figure across the x-axis. The x-coordinates remain unchanged, while the y-coordinates become their opposites. This transformation preserves the shape and size of the figure, making the original and reflected triangles congruent.

5 Triangle Emma with vertices E(5,10), M(10,10), M(5,15) is reflected over the x-axis. What are the coordinates of E'?

Hint: When reflecting over the x-axis, the x-coordinate stays the same while the y-coordinate changes sign.

Show the answer

Answer: (5,-10)

  1. Identify the original coordinates of point E: (5,10)
  2. Apply reflection over the x-axis - keep x-coordinate the same, change sign of y-coordinate
  3. x-coordinate remains: 5
  4. y-coordinate changes from 10 to -10
  5. The reflected point E' is (5,-10)

6 Triangle Aroha with vertices A(3,5), B(7,5), C(5,9) is reflected over the x-axis. What are the coordinates of A'B'C'?

Hint: When reflecting over the x-axis, the x-coordinates stay the same while the y-coordinates change their sign.

Show the answer

Answer: (3,-5),(7,-5),(5,-9)

  1. Original vertices are A(3,5), B(7,5), C(5,9)
  2. Reflection over x-axis: x-coordinates remain the same, y-coordinates change sign
  3. A' = (3,-5)
  4. B' = (7,-5)
  5. C' = (5,-9)
  6. The coordinates of A'B'C' are (3,-5), (7,-5), (5,-9)
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