You are watching: How to reflect over a diagonal line

Suppose you have a rectangle where the line through the diagonal is a line of symmeattempt. Using what you know about reflections, define why the rectangle have to be a square.

## IM Commentary

The goal of this job is to give students experience applying and reasoning around reflections of geometric numbers utilizing their flourishing knowledge of the properties of rigid motions. In the case of reflecting a rectangle over a diagonal, the reflected image is still a rectangle and it shares 2 vertices via the original rectangle. The examples show that the reflected picture looks favor a rotation of the original rectangle around its center suggest. Furthermore, this "displacement" of the rectangle becomes smaller and also smaller sized as the rectangle becomes closer to being a square. This leads normally to the last part of the question: lines via a diagonal are lines of symmeattempt for a square (but not a non-square rectangle). Keep in mind that a line of symmeattempt can be believed of as a change that mirrors a figure is congruent to itself in a non-trivial method.

The job is intfinished for instructional objectives and also assumes that students recognize the properties of rigid revolutions defined in 8.G.1. Keep in mind that the vertices of the rectangles in question carry out not autumn exactly at intersections of the horizontal and also vertical lines on the grid. This suggests that students have to approximate and this gives an extran obstacle. Also offering a challenge is the fact that the grids have actually been attracted so that they are aligned with the diagonal of the rectangles fairly than being aligned with the vertical and also horizontal directions of the page. However before, this option of grid also renders it less complicated to factor around the reflections if they understand the descriptions of rigid movements shown in 8.G.3.

Keep in mind that students research lines of symmeattempt in 4th grade, but only informally. While the 8th grade requirements perform not need students to factor as formally as they will in high college geometry, they are certainly able to reason even more formally than they did in fourth grade. The solution to this task mirrors this expectation. See4.G Lines of symmetry for quadrilateralsfor an example of what students might be supposed to execute in fourth grade.

## Solution

The reflections of each rectangle are pictured listed below in blue. In each casethe reflected image shares 2 vertices with the original rectangle. This is bereason the line of reflection passes via two vertices and reflection over a line leaves all points on the line in their original place.

Notice that the reflected rectangle is, in each situation, still a rectangle ofthe very same dimension and also shape as the original rectangle. Also notice that as the size and also width of the rectangles end up being closer to one one more, the 2 vertices are getting closer and closer to the vertices of the original rectangle. The instance where the length and width of the rectangleare equal is examined in component (b).

For each of the photos listed below, the exact same type of reasoning applies: the redline deserve to be believed of as the $x$-axis of the grid. Reflecting over the $x$-axis does not readjust the $x$-coordinate of a allude however it transforms the sign of the $y$-coordinate. In the photo for (i), the rectangle vertex above the red line is just under 3 boxes over the red line. So its reflection is just under 3 systems below the red line, through the very same $x$ coordinate. Similarly in the photo for component (iii), the vertex of the rectangle below the red line is a small more than $3$ boxes listed below the red line: so its reflection will be on the same line via the $x$-axis but a small over three boxes * above* the red line. Comparable thinking applies to all vertices in the three pictures. Due to the fact that reflections map line segments to line segments, understanding wbelow the vertices of the rectangles map is sufficient to recognize the reflected image of the rectangle.

Below is a picture of a quadrilateral $PQRS$ through the line $overleftrightarrowPS$ containing the diagonal colored red. We assume $PQRS$is mapped to itself by reflection over the $overleftrightarrowPS$.

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Since $P$ and also $S$ are on the line of reflection, they are mapped to themselvesby the reflection. So the truth that the rectangle is mapped to itself indicates that $Q$maps to $R$ and $R$ to $Q$. Because opposite sides of a rectangle are congruent we know that $|RS| = |PQ|$ and also $|PR| = |QS|$. Reflection around line $overleftrightarrowPS$ preserves line segments and also sends segment $overlinePR$ to segment $overlinePQ$. Reflections preserve line segment lengths so $|PQ| = |PR|$. Putting our equalities together we find$$|RS| = |PQ| = |PR| = |QS|.$$Due to the fact that $PQRS$ is a rectangle whose 4 sides have actually the exact same length, it need to be a square.