How to Solve: Reflection and Symmetry

How to Solve: Reflection and Symmetry

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you can’t reflect a shape perfectly in 30 seconds, you’re losing easy marks on symmetry questions—let’s fix that now."

What You Need To Know First

1. Coordinate plane basics – You must know how to plot points (x, y) and read coordinates.
2. Basic shapes – Recognize triangles, quadrilaterals, and regular polygons (e.g., squares, equilateral triangles).
3. Perpendicular lines – Understand that a mirror line is perpendicular to the line joining a point and its image.

Key Vocabulary

Formulas To Know

1. Reflection Over the x-axis

Formula: (x, y) → (x, –y) - x = x-coordinate (stays the same) - y = y-coordinate (sign flips) Memorise This.

2. Reflection Over the y-axis

Formula: (x, y) → (–x, y) - x = x-coordinate (sign flips) - y = y-coordinate (stays the same) Memorise This.

3. Reflection Over the Line y = x

Formula: (x, y) → (y, x) - Swap x and y coordinates. Memorise This.

4. Reflection Over the Line y = –x

Formula: (x, y) → (–y, –x) - Swap x and y, then flip both signs. Memorise This.

5. Reflection Over a Vertical or Horizontal Line (e.g., x = a or y = b)

Method: - For x = a: Count how far the point is from the line, then place the image the same distance on the other side. - For y = b: Same as above, but vertically. Given on exam sheet (or memorise the logic).

6. Number of Lines of Symmetry in Regular Polygons

Formula: Number of sides = Number of lines of symmetry - Example: A regular pentagon (5 sides) has 5 lines of symmetry. Memorise This.

Step-by-Step Method

How to Reflect a Point Over a Mirror Line

Step 1: Identify the mirror line (e.g., x-axis, y-axis, y = x, or a custom line like x = 2). Step 2: Measure the perpendicular distance from the point to the mirror line. Step 3: Plot the image the same distance on the opposite side of the mirror line. Step 4: Write the coordinates of the image.

For shapes: - Reflect each vertex (corner) one by one. - Connect the reflected points in the same order.

Worked Example (Using Steps)

Question: Reflect point A(3, 4) over the line y = 1.

Step 1: Mirror line is y = 1 (horizontal line). Step 2: Distance from A to y = 1: - A’s y-coordinate = 4 - Mirror line y = 1 - Distance = |4 – 1| = 3 units Step 3: Image A’ must be 3 units below y = 1. - New y-coordinate = 1 – 3 = –2 - x-coordinate stays the same (3). Step 4: A’ = (3, –2)

Answer: The reflected point is (3, –2).

Worked Examples

Example 1 – Basic: Reflect Over the x-axis

Question: Reflect triangle ABC with vertices A(2, 3), B(4, 1), C(5, 5) over the x-axis.

Working: 1. Apply (x, y) → (x, –y) to each point.
- A(2, 3) → A’(2, –3)
- B(4, 1) → B’(4, –1)
- C(5, 5) → C’(5, –5) 2. Plot A’, B’, C’ and connect them.

Answer: A’(2, –3), B’(4, –1), C’(5, –5)

What we did and why: - Used the x-axis reflection rule to flip y-coordinates. - Applied the rule to every vertex to get the full reflected shape.

Example 2 – Medium: Reflect Over y = x

Question: Reflect quadrilateral DEFG with vertices D(1, 2), E(3, 4), F(5, 2), G(3, 0) over the line y = x.

Working: 1. Apply (x, y) → (y, x) to each point.
- D(1, 2) → D’(2, 1)
- E(3, 4) → E’(4, 3)
- F(5, 2) → F’(2, 5)
- G(3, 0) → G’(0, 3) 2. Plot D’, E’, F’, G’ and connect them.

Answer: D’(2, 1), E’(4, 3), F’(2, 5), G’(0, 3)

What we did and why: - Swapped x and y for each point (y = x reflection rule). - Ensured the shape’s order was preserved to avoid twisting.

Example 3 – Exam Style: Disguised Reflection

Question: A shape has vertices P(–2, 1), Q(0, 3), R(2, 1). After reflection, P’ is at (1, –2). Find the mirror line.

Working: 1. Find the midpoint of PP’ (mirror line passes through it).
- P(–2, 1), P’(1, –2)
- Midpoint = ((–2 + 1)/2, (1 + –2)/2) = (–0.5, –0.5) 2. Find the slope of PP’.
- Slope = (–2 – 1)/(1 – –2) = –3/3 = –1 3. Mirror line is perpendicular to PP’ (slope = 1, since –1 × 1 = –1). 4. Use point-slope form with midpoint (–0.5, –0.5):
- y – (–0.5) = 1(x – (–0.5))
- y + 0.5 = x + 0.5
- y = x

Answer: The mirror line is y = x.

What we did and why: - Used the property that the mirror line is the perpendicular bisector of the line joining a point and its image. - Calculated midpoint and slope to derive the mirror line equation.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in. Reflection is just a flip over a mirror line. Here’s what you must remember: 1. x-axis? Keep x, flip y: (x, y) → (x, –y). 2. y-axis? Flip x, keep y: (x, y) → (–x, y). 3. y = x? Swap x and y: (x, y) → (y, x). 4. y = –x? Swap and flip both: (x, y) → (–y, –x). 5. Custom line? Measure perpendicular distance, plot the same distance on the other side.

For symmetry, count lines by folding—if it matches, it’s a line of symmetry. Regular polygons? Number of sides = number of lines. Irregular? Check each one.

Now go practice—reflect 3 points, find a mirror line, and count symmetry lines. You’ve got this!