KCSE 2025 Maths P1 Q13 — Reflection and the Mirror Line
Published
The Question
“Triangle ABC has been reflected in a mirror line to give an image. On the grid you are given the object triangle ABC together with two image points, A dash and C dash. Draw the mirror line and hence complete the image triangle A dash B dash C dash. (3 marks)”
Recall what a mirror line is
A reflection sends every point to an image that sits directly opposite it across the mirror, the same distance away. Because of this, the mirror line is exactly the perpendicular bisector of the segment joining any point to its image. This single idea is the key to the whole question, so we use it to locate the line.
Join a point to its image and bisect it
Take the object point A and its image A dash and draw the straight segment between them. Find the midpoint of this segment, then draw a line through that midpoint that meets the segment at a right angle. That perpendicular bisector is your mirror line.
Confirm the line using the second pair
You were given a second image point, C dash, so use it as a check. The mirror line you drew should also pass through the midpoint of C to C dash and cross it at a right angle. When it does, you know the line is correct and not just an accident of one pair of points.
Using a second pair of corresponding points is a built-in check that costs no extra marks but guards against error.
Reflect B to find its image
The image of the third vertex, B dash, is not given, so construct it. From B, drop a line to the mirror that meets it at a right angle, measure that distance, then continue the same distance on the far side of the line. The point you reach is B dash.
Complete the image triangle
Now that all three image vertices are known, join A dash to B dash, B dash to C dash, and C dash back to A dash. That finished triangle is the required image of ABC.
Final Result
The mirror line is the perpendicular bisector of A to A dash (and equally of C to C dash). Reflecting B across this line gives B dash, and joining A dash, B dash and C dash completes the image triangle.
Why this method works
Reflection preserves distance from the mirror: an object point and its image are the same perpendicular distance from the line, one on each side. So the mirror must pass through the midpoint of the segment joining a point to its image and must be perpendicular to it — that is precisely the definition of a perpendicular bisector. Since every corresponding pair of points shares this same mirror, any one pair fixes the line and any other pair confirms it, which is why the same construction both finds and verifies the answer.
The mirror line meets both AA dash and CC dash at their midpoints at 90 degrees, and B dash lies the same distance from the line as B on the opposite side.