Transformations

Grade 9 · geometry · 44 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 44, worked through step by step — try them before opening the answer.

1 Triangle ABC with vertices A(2, 4), B(6, 4), C(4, 8) is reflected across the x-axis, then translated 4 units right. Find the coordinates of A''B''C''.

Hint: First apply the reflection transformation to all points, then apply the translation to the reflected points. Remember that reflection across the x-axis changes the sign of the y-coordinate.

Show the answer

Answer: A''(6, -4), B''(10, -4), C''(8, -8)

When performing composite transformations, apply them in sequence. Reflection across the x-axis preserves x-coordinates but negates y-coordinates. Translation moves all points by the same horizontal and/or vertical displacement.

2 Triangle ABC with vertices A(2, 4), B(6, 4), C(4, 8) is reflected across the y-axis, then translated 4 units right. Find the coordinates of A''B''C''.

Hint: First apply the reflection transformation to each vertex, then apply the translation to the reflected coordinates.

Show the answer

Answer: A''(2, 4), B''(-2, 4), C''(0, 8)

When reflecting across the y-axis, the x-coordinate changes sign while the y-coordinate stays the same. A translation moves all points by the same amount in the specified direction.

3 Triangle ABC with vertices A(5, 0), B(10, 5), C(5, 10) is reflected across the x-axis, then rotated 90° counterclockwise about the origin. Find the coordinates of A''B''C''.

Hint: Remember that reflection across the x-axis changes the sign of y-coordinates, and a 90° counterclockwise rotation about the origin transforms (x, y) to (-y, x).

Show the answer

Answer: A''(0, -5), B''(-5, -10), C''(-10, -5)

When performing multiple transformations, apply them in sequence. Reflection flips the figure across a line, while rotation turns it around a point. The order of transformations matters in composite transformations.

4 Triangle ABC with vertices A(4, 6), B(8, 6), C(6, 10) is reflected across the line x = 2, then rotated 90° counterclockwise about the origin. Find the coordinates of A''B''C''.

Hint: First apply the reflection transformation, then apply the rotation transformation to the reflected coordinates. Remember that reflection across a vertical line changes x-coordinates, and 90° counterclockwise rotation follows the rule (x, y) → (-y, x).

Show the answer

Answer: A''(-6,0), B''(-6,-4), C''(-10,-2)

  1. Reflect across the line x = 2. The distance from a point to the line x = 2 is |x - 2|. After reflection, the new x-coordinate becomes 2 - (x - 2) = 4 - x. A(4, 6) → A'(4 - 4, 6) = A'(0, 6) B(8, 6) → B'(4 - 8, 6) = B'(-4, 6) C(6, 10) → C'(4 - 6, 10) = C'(-2, 10)
  2. Rotate 90° counterclockwise about the origin using the rule (x, y) → (-y, x). A'(0, 6) → A''(-6, 0) B'(-4, 6) → B''(-6, -4) C'(-2, 10) → C''(-10, -2) The final coordinates are A''(-6, 0), B''(-6, -4), C''(-10, -2).

5 Triangle Emma with vertices E(5,10), M(10,15), M(15,10) is reflected over the y-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final image?

Hint: Remember that reflection over the y-axis changes the sign of x-coordinates, and rotation 90° counterclockwise about the origin transforms (x,y) to (-y,x).

Show the answer

Answer: E'(-10,-5), M'(-15,-10), M'(-10,-15)

When performing multiple transformations, apply them in sequence. Reflection over the y-axis preserves y-values but negates x-values. A 90° counterclockwise rotation about the origin swaps coordinates and changes signs in a specific pattern.

6 Triangle Emma with vertices E(1,3), M(5,7), M(9,3) is reflected over the x-axis, then rotated 90° counterclockwise about the origin. Find the coordinates of the final image of vertex E.

Hint: Remember that reflection over the x-axis changes the sign of the y-coordinate, and a 90° counterclockwise rotation about the origin transforms (x,y) to (-y,x).

Show the answer

Answer: (-3,-1)

When performing composite transformations, apply them in sequence from right to left. Reflection over the x-axis preserves x-coordinates but negates y-coordinates. A 90° counterclockwise rotation about the origin swaps coordinates and changes signs according to a specific pattern. Try applying these transformations to a different point like (2,4) to understand the pattern.

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Also taught in Grade 8