Reflections

A fold defines the image

A reflection sends a point across a chosen line to the place that matches it when the plane is folded along that line. For example, a dot $3$ cm from a straight crease matches a dot $3$ cm away on the opposite side, measured along the same perpendicular.

The line is the mirror line, also called the line of reflection. As in Translations, write $P\mapsto P'$ for a point and its image. A reflection uses one mirror for the whole plane. The segment joining a point to its image is its connector.

Here is the exact reflection definition. If $P$ is off mirror $m$, its image $P'$ is on the opposite side, the segment $PP'$ is perpendicular to $m$, and their intersection $H$ satisfies $PH=HP'$. Thus $m$ is the perpendicular bisector of $PP'$: it passes through its midpoint at a right angle. This describes the reflection pair; a general theorem about all points equidistant from two endpoints comes in lesson 9.1.

If $P$ lies on $m$, define $P'=P$. Its connector has zero length, so there is no nonzero segment to mark perpendicular. Do not add a right-angle mark to this case.

Equal distances to a line alone do not locate an image. The two points must also line up along one perpendicular. Sliding a proposed image along the far side could keep its distance to the mirror unchanged while losing that alignment.

Which points stay fixed?

A fixed point is a point whose image is itself. Reflect a point on a crease and it stays in place; reflect one $2$ cm off the crease and it goes to the other side. The definition therefore tells us that the fixed points of a reflection are exactly the points on its mirror.

This is a point-by-point claim. For example, reflect the entire horizontal axis $y=0$ across the vertical axis $x=0$. The horizontal axis occupies the same set of positions afterward, but its point $(3,0)$ moves to $(-3,0)$. Only its crossing with the mirror is fixed. A set can occupy its old position while its points exchange places. Testing when a whole figure matches itself will be the focus of Symmetry.

Horizontal and vertical mirrors

Across a vertical mirror, a point keeps its height. Suppose the mirror is $x=2$ and a point is $(6,1)$. The point is $4$ units to the right of the mirror. Its image must be $4$ units left of it, at $(-2,1)$.

The image’s horizontal coordinate and the starting horizontal coordinate are equally spaced around $2$. Their sum is $4$, twice the mirror coordinate. For a vertical mirror $x=h$, this becomes $x+x'=2h$, so

$x'=2h-x.$

Across a horizontal mirror, a point keeps its horizontal position. A point $(2,4)$ is $5$ units above $y=-1$, so its image is $5$ units below, at $(2,-6)$. In general, for $y=k$, the heights satisfy $y+y'=2k$.

The axes are the zero-coordinate cases of the first two rules. Reflecting across the $x$-axis changes the $y$-coordinate, because the distance across that mirror is vertical. A point on any of these mirrors satisfies the rule with no change.

The two diagonal mirrors

A diagonal fold can exchange horizontal and vertical positions. Across $y=x$, the point $(3,1)$ matches $(1,3)$. Folding the square grid along that diagonal exchanges its horizontal and vertical directions with their positive directions matched. The horizontal coordinate becomes the vertical coordinate and vice versa, including their signs:

$\text{Across }y=x:\quad(x,y)\mapsto(y,x).$

You met this exchange when graphing inverse functions in Algebra I. Here it applies to every point of the plane, including points of figures that are not function graphs.

For the other diagonal, $y=-x$, the fold matches the positive horizontal direction with the negative vertical direction. For example, $(3,1)$ matches $(-1,-3)$. The roles exchange and both signs reverse:

$\text{Across }y=-x:\quad(x,y)\mapsto(-y,-x).$

These are different mirrors. A point $(a,a)$ lies on $y=x$ and stays fixed there. A point $(a,-a)$ lies on $y=-x$ and stays fixed there. Both rules fix the origin.

What the fold preserves and reverses

A reflection preserves the distances within a figure and its angle openings. Imagine tracing a $6$ cm segment and a $72^\circ$ angle onto one side of thin paper, then folding and tracing them onto the other side. The tracing matches the segment without stretching it and the rays without opening or closing their angle.

We accept reflection preserves every distance and every angle measure as a geometric foundation directly from this fold picture. Reflection is therefore a rigid motion. Measuring a few traced examples supplies evidence; it is not a proof about all points. The later equidistance proof belongs to Perpendicular Bisectors and the Circumcenter, lesson 9.1.

What changes is orientation, the clockwise or counterclockwise order of labeled points around a noncollinear triangle. A slide does not turn the marked face over. From that slide picture, we accept that a translation preserves this clockwise or counterclockwise order. A fold turns the marked face over, reversing this order. We accept orientation reversal directly from that fold picture. A single point or a straight segment has no clockwise boundary order to reverse.

You can draw a reflection across an oblique line on paper without a coordinate rule. With a ruler and protractor, draw a perpendicular through each off-mirror vertex and locate its foot. Use a compass to copy that vertex-to-foot distance to the opposite side along the same perpendicular. Keep any vertex on the mirror unchanged, then join the image vertices in corresponding order. This uses the angle-drawing foundation and the length-copy method already taught; pencil measurements are approximate. It does not assume a later compass-and-straightedge perpendicular construction.

Common mistakes