How to Solve: Transformations

How to Solve: Transformations

(Geometry – For Students & Teachers)

Introduction

"Imagine you’re designing a video game level—how do you make a character flip, slide, or spin without breaking the rules? That’s transformations. Master this, and you’ll ace every geometry exam question on reflections, rotations, translations, and dilations!

What You Need To Know First

Before diving into transformations, ensure you understand: 1. Coordinate Plane Basics – Plotting points, reading (x, y) coordinates, and understanding quadrants. 2. Basic Shapes & Properties – Recognizing triangles, rectangles, and other polygons. 3. Distance & Midpoint Formulas – Used in reflections and rotations (given on most exam sheets).

Key Vocabulary

Formulas To Know

1. Translation (Slide)

Formula: (x, y) → (x + a, y + b) - x, y = Original coordinates - a = Horizontal shift (right if +, left if –) - b = Vertical shift (up if +, down if –) Memorise This.

Example: Translate (2, 3) by (4, -1) → (2 + 4, 3 + (-1)) = (6, 2)

2. Reflection (Flip)

Formulas (given on exam sheet, but memorize for speed): - Over x-axis: (x, y) → (x, -y) - Over y-axis: (x, y) → (-x, y) - Over line y = x: (x, y) → (y, x) - Over line y = -x: (x, y) → (-y, -x)

MEMORISE THE FIRST TWO (x-axis and y-axis).

Example: Reflect (3, 5) over the x-axis → (3, -5)

3. Rotation (Turn)

Formulas (given on exam sheet, but memorize 90° and 180°): - 90° clockwise: (x, y) → (y, -x) - 90° counterclockwise: (x, y) → (-y, x) - 180° (either way): (x, y) → (-x, -y)

MEMORISE 90° AND 180° ROTATIONS.

Example: Rotate (2, 4) 90° clockwise → (4, -2)

4. Dilation (Resize)

Formula: (x, y) → (kx, ky) - k = Scale factor (enlarge if k > 1, shrink if 0 < k < 1) - Center of dilation = Fixed point (usually origin unless stated).

Memorise This.

Example: Dilate (3, 6) by scale factor 2 → (6, 12)

Step-by-Step Method

How to Solve Any Transformation Problem

Step 1: Identify the transformation type. - Is it a slide (translation), flip (reflection), turn (rotation), or resize (dilation)?

Step 2: Write down the rule for that transformation. - Use the formulas above.

Step 3: Apply the rule to every vertex (corner) of the shape. - If given a single point, apply it once. - If given a shape, apply it to all vertices.

Step 4: Plot the new points (if required). - Connect the dots to draw the transformed shape.

Step 5: Check for isometry (if needed). - Did the size stay the same? (Yes for translations, rotations, reflections. No for dilations.)

Worked Example (Using Steps)

Problem: Triangle ABC has vertices A(1, 2), B(3, 4), and C(5, 1). Translate it by (2, -3). Find the new coordinates.

Solution: 1. Identify: Translation (slide). 2. Rule: (x, y) → (x + 2, y - 3) 3. Apply to each vertex:
- A(1, 2) → (1 + 2, 2 - 3) = A’(3, -1)
- B(3, 4) → (3 + 2, 4 - 3) = B’(5, 1)
- C(5, 1) → (5 + 2, 1 - 3) = C’(7, -2) 4. Plot (if needed): Draw the new triangle A’B’C’. 5. Check: Size is the same (isometry).

Answer: A’(3, -1), B’(5, 1), C’(7, -2)

Worked Examples

Example 1 – Basic (Reflection)

Problem: Reflect point P(4, -2) over the y-axis. Find P’.

Solution: 1. Identify: Reflection over y-axis. 2. Rule: (x, y) → (-x, y) 3. Apply: (4, -2) → (-4, -2) 4. Check: Distance from y-axis flips sign.

Answer: P’(-4, -2)

What we did and why: We used the y-axis reflection rule because the problem asked for it. The y-coordinate stays the same, but the x-coordinate changes sign.

Example 2 – Medium (Rotation)

Problem: Rotate point Q(-1, 3) 90° counterclockwise about the origin. Find Q’.

Solution: 1. Identify: 90° counterclockwise rotation. 2. Rule: (x, y) → (-y, x) 3. Apply: (-1, 3) → (-3, -1) 4. Check: Plot to confirm the turn.

Answer: Q’(-3, -1)

What we did and why: We used the 90° counterclockwise rule because the problem specified direction. The x and y swap, and the new x becomes negative.

Example 3 – Exam Style (Dilation + Translation)

Problem: A square has vertices at (1, 1), (1, 3), (3, 3), and (3, 1). First, dilate it by a scale factor of 2 centered at the origin. Then, translate it by (-1, 4). Find the final coordinates.

Solution: 1. First transformation: Dilation (k = 2)
- (1, 1) → (2, 2)
- (1, 3) → (2, 6)
- (3, 3) → (6, 6)
- (3, 1) → (6, 2)

1. Second transformation: Translation (-1, 4)
2. (2, 2) → (2 - 1, 2 + 4) = (1, 6)
3. (2, 6) → (1, 10)
4. (6, 6) → (5, 10)
5. (6, 2) → (5, 6)

Answer: (1, 6), (1, 10), (5, 10), (5, 6)

What we did and why: We did one transformation at a time—first dilation (resizing), then translation (sliding). Order matters! If we translated first, the dilation would stretch the translation too.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, exam’s tomorrow—here’s your 60-second crash course on transformations!

1. Translations? Just add/subtract to x and y. (x, y) → (x + a, y + b)
2. Reflections? Flip over x-axis → y changes sign. Flip over y-axis → x changes sign.
3. Rotations? 90° clockwise = (y, -x). 90° counterclockwise = (-y, x). 180° = (-x, -y).
4. Dilations? Multiply x and y by the scale factor. (x, y) → (kx, ky)
5. Order matters! Dilation first, then translation. Not the other way around.
6. Check your work: Did you apply the rule to every vertex? Did you mix up x and y?

You’ve got this. Now go ace that exam!