Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 1: Transformations

Lesson 3: Reflections

In this Grade 7 lesson from Big Ideas Math, Course 2 Accelerated, students learn how to identify reflections and reflect figures across the x-axis and y-axis in the coordinate plane. Students explore how reflections produce congruent mirror images by analyzing frieze patterns — repeating horizontal designs — to determine whether they reflect onto themselves horizontally or vertically. The lesson connects geometric transformations to real-world contexts while building understanding of symmetry, congruence, and coordinate geometry aligned to standards 8.G.1–8.G.3.

Section 1

Reflections

Property

A reflection is a motion that leaves every point on a line LL (the line of reflection) fixed, and for a point PP not on LL, with P′P' its image, LL is the perpendicular bisector of the line segment PP′PP'.

  • A reflection preserves the lengths of line segments and the measures of angles.
  • A reflection leaves every point on the line of reflection fixed and interchanges the two sides of that line.
  • A reflection reverses orientation.

Examples

  • The reflection of the point P(4,7)P(4, 7) across the xx-axis is the point P′(4,−7)P'(4, -7). The xx-coordinate stays the same, and the yy-coordinate changes sign.
  • The reflection of the point Q(−2,5)Q(-2, 5) across the yy-axis is the point Q′(2,5)Q'(2, 5). The yy-coordinate stays the same, and the xx-coordinate changes sign.
  • The reflection of the point R(3,8)R(3, 8) across the line y=xy=x is the point R′(8,3)R'(8, 3). The coordinates are interchanged.

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

Identifying Reflectional Symmetry in Frieze Patterns

Property

A frieze pattern has reflectional symmetry if there exists a vertical line of reflection that divides the pattern so that one half is the mirror image of the other half. The line of reflection acts as a "mirror" where corresponding points are equidistant from the line but on opposite sides.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Reflections

Property

A reflection is a motion that leaves every point on a line LL (the line of reflection) fixed, and for a point PP not on LL, with P′P' its image, LL is the perpendicular bisector of the line segment PP′PP'.

  • A reflection preserves the lengths of line segments and the measures of angles.
  • A reflection leaves every point on the line of reflection fixed and interchanges the two sides of that line.
  • A reflection reverses orientation.

Examples

  • The reflection of the point P(4,7)P(4, 7) across the xx-axis is the point P′(4,−7)P'(4, -7). The xx-coordinate stays the same, and the yy-coordinate changes sign.
  • The reflection of the point Q(−2,5)Q(-2, 5) across the yy-axis is the point Q′(2,5)Q'(2, 5). The yy-coordinate stays the same, and the xx-coordinate changes sign.
  • The reflection of the point R(3,8)R(3, 8) across the line y=xy=x is the point R′(8,3)R'(8, 3). The coordinates are interchanged.

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

Identifying Reflectional Symmetry in Frieze Patterns

Property

A frieze pattern has reflectional symmetry if there exists a vertical line of reflection that divides the pattern so that one half is the mirror image of the other half. The line of reflection acts as a "mirror" where corresponding points are equidistant from the line but on opposite sides.

Examples