A reflection creates a mirror image of a figure across a line. Every point of the original figure lands the same distance from that line, on the opposite side of it, and the mirror image is congruent (equal in measure) to the original — only its orientation is reversed. This page works through the reflections used most often, then shows how the same idea reflects a point across any line at all.
Transformation. A transformation is a function that moves or changes a figure in some way to produce a new image. The figure before a transformation is applied to it is the pre-image, and the figure that results is its image, also called the post-image. The image of point P is written P′ (read “P prime”). The terms are relative to the step being considered: a figure that is the image of one step can be the pre-image of the next.
Reflection. A reflection is a transformation that creates a mirror image of a figure about a specified line. The pre-image and the image are equidistant from the line of reflection: each point and its image are the same distance from the line, on opposite sides of it.
Line of reflection. The line of reflection is the line across which a figure is reflected. It is the perpendicular bisector of the segment joining any point of the pre-image to its image, so it intersects that segment at its midpoint and at a right angle.
Rigid motion. A rigid motion is any transformation that maintains congruence. A reflection is a rigid motion — distances and angle measures do not change, and the image is congruent (equal in measure) to the pre-image — but it reverses the orientation of the figure.
Composition. A composition is two or more transformations combined and applied in a specific order, such as a translation followed by a reflection. The image of one step becomes the pre-image of the next step — it is after the first step and before the second.
What you are looking at: a triangle on a coordinate grid, its mirror image, and a line of reflection. This activity uses the standard lines only: the x-axis, the y-axis, a horizontal line y = c, a vertical line x = c, and the lines y = x and y = -x.
What to do: choose a line of reflection. For the lines y = c and x = c, use the stepper to set c. Then drag any vertex of the blue triangle to a new grid point and watch the image follow. Choose a vertex (A, B, or C) to see the segment from that vertex to its image: the marked intersection point is where that segment intersects the line of reflection, and it is the midpoint of the segment.
What to notice: the line of reflection is the perpendicular bisector of the segment from each point to its image, so each point and its image are exactly the same distance from the line. The image is congruent (equal in measure) to the pre-image, with its orientation reversed.
What you are looking at: a point, or one vertex of a triangle, together with a standard line of reflection. The image is not drawn.
What to do: work out the image coordinates in your head or on paper first, then choose your answer. A first wrong answer earns a hint and one retry. After a second miss, the correct answer is named and explained.
What to notice: each rule changes only what the line requires — a reflection in a horizontal line changes only the y-coordinate, a reflection in a vertical line changes only the x-coordinate, and the lines y = x and y = -x swap the coordinates (with signs changed for y = -x).
What you are looking at: a blue pre-image triangle and a dashed target triangle. Your job is to find the reflection that sends the pre-image exactly onto the target.
What to do: select one of the offered standard lines, then press Apply Reflection. If the image lands exactly on the target, the item is complete. A first wrong line earns a hint and one retry; after a second miss, the correct line is named, explained, and applied for you. The later items are compositions in two steps: first translate the triangle by the given rule, then reflect the result in one of the offered lines.
What to notice: only one offered line can be correct. Check it vertex by vertex: the correct line is the perpendicular bisector of the segment from every vertex to its target image.
For advanced students: this activity goes beyond the standard lines used in the rest of this page. It reflects a single point across a slanted line y = mx + b, where none of the ready-made coordinate rules applies. It follows the method from the notes, in four steps.
What to do: work through the four steps in order. Step 1: find the slope of the given line. Step 2: find the equation of the line through the point perpendicular to the given line — its slope is the opposite reciprocal. Step 3: find the intersection point of the two lines. Step 4: use that the intersection point is the midpoint of the segment from the point to its image, and find the image point. A first wrong answer at any step earns a hint and one retry; after a second miss, the answer to that step is named and explained.
What to notice: the method is the perpendicular bisector idea from the Explorer, used in reverse — the line of reflection must bisect the segment from the point to its image at a right angle, and that fact is enough to locate the image exactly.
Six questions, one at a time. A first wrong answer earns a hint and one retry; the full explanation appears when a question is answered correctly or after a second miss. Your score counts first-try answers only.
Lesson 33 · Reflections · Mr. J Geometry. All activities on this page run entirely in your browser.