Coordinate Rules: Reflection Across Other Lines (y = k or x = h)
Coordinate Rules: Reflection Across Other Lines (y = k or x = h) is a Grade 7 math skill in Reveal Math Accelerated, Unit 6: Congruence and Similarity, where students apply coordinate rules to reflect figures across horizontal lines y = k and vertical lines x = h that are not the axes, using the relationship between a point and the line of reflection to find the image coordinates. This extends basic axis reflections to any horizontal or vertical line.
Key Concepts
Property Sometimes the mirror isn't the main axis, but a different horizontal line (like $y = 2$) or vertical line (like $x = 1$). For a horizontal line $y = k$: The x coordinate stays the same. The new y is calculated using $(x, y) \rightarrow (x, 2k y)$. For a vertical line $x = h$: The y coordinate stays the same. The new x is calculated using $(x, y) \rightarrow (2h x, y)$.
Examples Reflect point (3, 5) across the horizontal line $y = 2$: The x coordinate stays 3. The formula for y is $2(k) y$. Here $k = 2$ and $y = 5$. $2(2) 5 = 4 5 = 1$. The reflected point is (3, 1). Alternative Counting Method for the same point: From y=5 down to the mirror y=2 is a distance of 3 units. Go 3 more units down from the mirror (2 3 = 1). The new y is 1.
Explanation This is the most challenging part of reflections! The biggest mistake students make is confusing the lines. They see $y = 2$ and think "y axis". Remember: $y = 2$ is a horizontal line where all y values are 2. If the formula $2k y$ feels confusing, do not panic. Always sketch the line on your graph paper and use the "Counting Method" from Session 2. Counting distance to the mirror and out the other side works 100% of the time, no matter where the mirror is placed.
Common Questions
How do you reflect a point across the line y = k?
The y-coordinate of the image is found by reflecting the point vertically across y = k. The new y-coordinate is 2k - y_original. The x-coordinate stays the same.
How do you reflect a point across the line x = h?
The x-coordinate of the image is 2h - x_original. The y-coordinate stays the same.
What is the general strategy for reflections across lines parallel to the axes?
Find the distance from the point to the line, then place the image at the same distance on the opposite side. The coordinate perpendicular to the line changes; the parallel coordinate remains unchanged.
What is Reveal Math Accelerated Unit 6 about?
Unit 6 covers Congruence and Similarity, including all geometric transformations (translations, reflections, rotations, dilations), coordinate rules, and connections to congruence and similarity.