Section 2.3: Reflections

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.

Property

A reflection creates a mirror image of a figure across a line of reflection. To identify if two figures are reflections of each other, check if one figure can be flipped across a line to match the other exactly, with corresponding points equidistant from the line of reflection.

Examples

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

Property

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

• Across the X-Axis: The rule is $(x, y) \rightarrow (x, -y)$. The x-coordinate stays exactly the same, and the y-coordinate changes to its opposite sign.
• Across the Y-Axis: The rule is $(x, y) \rightarrow (-x, y)$. 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.