Learn on PengiReveal Math, AcceleratedUnit 6: Congruence and Similarity

Lesson 6-2: Explore Reflections

In this Grade 7 Reveal Math Accelerated lesson, students explore reflections as a type of rigid motion, learning that a reflection flips a figure across a line of reflection while preserving the size and shape of the original figure. Students practice identifying and drawing reflections on the coordinate plane, including reflections across the x-axis and y-axis, by analyzing how the coordinates of vertices change to produce the image. The lesson connects geometric concepts to real-world contexts such as furniture arrangement and symmetry in decorative art.

Section 1

Defining a Reflection

Property

A reflection is a rigid transformation that "flips" a figure across a specific line called the "line of reflection" (think of it as a mirror). Every point on the original figure (pre-image) has a matching point on the reflected figure (image). Because it is a rigid motion, the size and shape stay exactly the same (they are congruent). However, reflection is unique because it reverses the orientation—just like your left hand looks like a right hand in the mirror.

Examples

  • Macro View: A butterfly's wings, where the left wing is a perfect mirror image of the right wing across the center of its body.
  • Micro Detail (Distance): If point A is exactly 4 units away from the mirror line, its reflection A' will be exactly 4 units away on the opposite side.
  • Micro Detail (Perpendicular): If you draw a line connecting point A to A', that line will cross the mirror perfectly at a 90-degree angle.

Explanation

To truly master reflections, remember the "Mirror Rule". The line of reflection acts as the perfect halfway point (perpendicular bisector).A common mistake is thinking a reflection just "slides" the shape over the line. It doesn't! It flips it entirely. If the original triangle has a point pointing to the right, the reflected triangle's point will point to the left.

Section 2

Graphing Reflections by Counting

Property

Before memorizing any algebraic formulas, you can always reflect any point or polygon on a grid simply by counting. Find the perpendicular distance from an original vertex to the line of reflection, then count that exact same distance past the line to plot the new vertex. To reflect an entire shape, just repeat this counting process for each corner and connect the new dots.

Examples

  • Reflecting a Point: Point B is 3 grid squares above the x-axis. To reflect it across the x-axis, count 3 squares down to the axis, then 3 more squares down past the axis. Plot B'.
  • Reflecting a Polygon: To reflect triangle ABC, do not try to flip the whole triangle in your head!
    1. Count distance for A, plot A'.
    2. Count distance for B, plot B'.
    3. Count distance for C, plot C'.
    4. Connect A', B', and C'.

Explanation

Counting is your ultimate backup plan if you forget a formula.But beware of these two micro-traps:

  1. Don't count the dot you start on! Only count the jumps between grid lines.
  2. Count straight across! If your mirror line is vertical, you must count horizontally. If your mirror is horizontal, you must count vertically.

Section 3

Coordinate Rules: Reflection Across the X-Axis and Y-Axis

Property

When reflecting across the main coordinate axes, we use simple algebraic rules instead of counting.

  • Across the X-Axis: The rule is (x,y)→(x,−y)(x, y) \rightarrow (x, -y). The x-coordinate stays exactly the same, and the y-coordinate changes to its opposite sign.
  • Across the Y-Axis: The rule is (x,y)→(−x,y)(x, y) \rightarrow (-x, y). The y-coordinate stays exactly the same, and the x-coordinate changes to its opposite sign.
  • Memory Trick: "The axis you reflect across is the letter that stays the same!"

Examples

  • Reflect across X-axis (y changes): Point (3, 4) becomes (3, -4). Point (-2, -5) becomes (-2, 5).
  • Reflect across Y-axis (x changes): Point (3, 2) becomes (-3, 2). Point (-4, -1) becomes (4, -1).
  • Points ON the mirror: Reflect (0, 5) across the Y-axis. Since the rule says change the sign of x, -0 is still 0. The point stays at (0, 5) because it is already touching the mirror!

Explanation

Why does this math work?Imagine jumping over the horizontal x-axis. You are moving Up or Down. "Up and Down" is the y-direction! That is why the y-value flips (positive becomes negative, or negative becomes positive), while your left/right position (x) doesn't change at all. Always double-check your signs: changing a sign means if it was already negative, it becomes positive.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining a Reflection

Property

A reflection is a rigid transformation that "flips" a figure across a specific line called the "line of reflection" (think of it as a mirror). Every point on the original figure (pre-image) has a matching point on the reflected figure (image). Because it is a rigid motion, the size and shape stay exactly the same (they are congruent). However, reflection is unique because it reverses the orientation—just like your left hand looks like a right hand in the mirror.

Examples

  • Macro View: A butterfly's wings, where the left wing is a perfect mirror image of the right wing across the center of its body.
  • Micro Detail (Distance): If point A is exactly 4 units away from the mirror line, its reflection A' will be exactly 4 units away on the opposite side.
  • Micro Detail (Perpendicular): If you draw a line connecting point A to A', that line will cross the mirror perfectly at a 90-degree angle.

Explanation

To truly master reflections, remember the "Mirror Rule". The line of reflection acts as the perfect halfway point (perpendicular bisector).A common mistake is thinking a reflection just "slides" the shape over the line. It doesn't! It flips it entirely. If the original triangle has a point pointing to the right, the reflected triangle's point will point to the left.

Section 2

Graphing Reflections by Counting

Property

Before memorizing any algebraic formulas, you can always reflect any point or polygon on a grid simply by counting. Find the perpendicular distance from an original vertex to the line of reflection, then count that exact same distance past the line to plot the new vertex. To reflect an entire shape, just repeat this counting process for each corner and connect the new dots.

Examples

  • Reflecting a Point: Point B is 3 grid squares above the x-axis. To reflect it across the x-axis, count 3 squares down to the axis, then 3 more squares down past the axis. Plot B'.
  • Reflecting a Polygon: To reflect triangle ABC, do not try to flip the whole triangle in your head!
    1. Count distance for A, plot A'.
    2. Count distance for B, plot B'.
    3. Count distance for C, plot C'.
    4. Connect A', B', and C'.

Explanation

Counting is your ultimate backup plan if you forget a formula.But beware of these two micro-traps:

  1. Don't count the dot you start on! Only count the jumps between grid lines.
  2. Count straight across! If your mirror line is vertical, you must count horizontally. If your mirror is horizontal, you must count vertically.

Section 3

Coordinate Rules: Reflection Across the X-Axis and Y-Axis

Property

When reflecting across the main coordinate axes, we use simple algebraic rules instead of counting.

  • Across the X-Axis: The rule is (x,y)→(x,−y)(x, y) \rightarrow (x, -y). The x-coordinate stays exactly the same, and the y-coordinate changes to its opposite sign.
  • Across the Y-Axis: The rule is (x,y)→(−x,y)(x, y) \rightarrow (-x, y). The y-coordinate stays exactly the same, and the x-coordinate changes to its opposite sign.
  • Memory Trick: "The axis you reflect across is the letter that stays the same!"

Examples

  • Reflect across X-axis (y changes): Point (3, 4) becomes (3, -4). Point (-2, -5) becomes (-2, 5).
  • Reflect across Y-axis (x changes): Point (3, 2) becomes (-3, 2). Point (-4, -1) becomes (4, -1).
  • Points ON the mirror: Reflect (0, 5) across the Y-axis. Since the rule says change the sign of x, -0 is still 0. The point stays at (0, 5) because it is already touching the mirror!

Explanation

Why does this math work?Imagine jumping over the horizontal x-axis. You are moving Up or Down. "Up and Down" is the y-direction! That is why the y-value flips (positive becomes negative, or negative becomes positive), while your left/right position (x) doesn't change at all. Always double-check your signs: changing a sign means if it was already negative, it becomes positive.