Why reflections feel like magic in geometry
Ever watched a selfie camera flip your face like a mirror? That flip is a reflection, and the same idea works on a graph paper.
💡 In Simple Words: A reflection takes a point, line or shape and creates a twin on the other side of a mirror line. The distance from the mirror stays the same, just the side changes.
What does "reflection" really mean?
In geometry, a reflection is a transformation that flips every point over a specific line, called the line of reflection. Think of the line as a river; everything on one bank has a matching point directly across the water, at the same distance.
Reflecting a point across the most common lines
- x‑axis (the horizontal line where y=0): change the sign of the y‑coordinate, keep x the same.
- y‑axis (the vertical line where x=0): change the sign of the x‑coordinate, keep y the same.
- line y = x (the diagonal that climbs up right): swap the x and y values.
- line y = –x (the diagonal that falls down right): swap x and y, then change both signs.
Quick reference table
| Mirror line | Formula for image (x',y') |
|---|---|
| x‑axis (y=0) | (x, ‑y) |
| y‑axis (x=0) | (‑x, y) |
| y = x | (y, x) |
| y = –x | (‑y, ‑x) |
| General line ax+by+c=0 | Use perpendicular foot method (see steps below) |
Step‑by‑step method for any line
When the mirror line isn’t one of the easy ones, we follow a short recipe. It works for any straight line written as ax + by + c = 0 (that just means a, b and c are numbers that describe the line).
Here’s what each step does:
- Identify line: Decide which line you’re reflecting across.
- Write its equation: Put the line in the form ax+by+c=0 if it isn’t already.
- Find perpendicular slope: The slope (steepness) of a line that meets your mirror at a right angle is –a/b (the negative reciprocal of the original slope).
- Locate foot of perpendicular: The foot is the exact spot where a line drawn from the original point meets the mirror at a right angle. Solve the two equations together to get its coordinates (x₁, y₁).
- Use equal distance: The reflected point (x',y') is as far from the foot as the original point, just on the opposite side. So x' = 2x₁ – x and y' = 2y₁ – y.
Worked example: reflect (3, 4) across the line y = x + 1
Step 1: Write the line as ax+by+c=0. Rearrange y = x + 1 → x – y + 1 = 0, so a=1, b=‑1, c=1.
Step 2: Find the foot of the perpendicular. The perpendicular slope is the negative reciprocal of the original slope (original slope = 1, so perpendicular slope = –1). The line through (3,4) with slope –1 is y‑4 = –1(x‑3) → y = –x + 7.
Step 3: Solve the system:
- Original line: x – y + 1 = 0 → y = x + 1.
- Perpendicular line: y = –x + 7.
Let’s try a point not on the line: reflect (5, 2) across y = x + 1.
Perpendicular line through (5,2): y‑2 = –1(x‑5) → y = –x + 7.
Solve with y = x + 1:
- x + 1 = –x + 7 → 2x = 6 → x = 3.
- y = 3 + 1 = 4.
- x' = 2·3 – 5 = 1.
- y' = 2·4 – 2 = 6.
Quick revision bullets
- Reflection keeps distance from the mirror line unchanged.
- For x‑axis, flip the sign of y. For y‑axis, flip the sign of x.
- For y = x, swap coordinates. For y = –x, swap and change signs.
- General line: find foot of perpendicular, then use (x',y') = (2x₁‑x, 2y₁‑y).
- Always verify by checking that the original point, foot, and image are collinear and that the foot is the midpoint of the segment joining them.
📝 Likely Exam Questions
- Find the image of the point (‑2, 5) when reflected in the line y = –x.
Answer: Swap and change signs → (‑5, 2). - Reflect the triangle with vertices (1,2), (4,2), (1,5) in the x‑axis. Write the coordinates of the image triangle.
Answer: Change y‑sign: (1,‑2), (4,‑2), (1,‑5). - Find the image of (6,‑1) in the line 3x + 4y – 12 = 0.
Answer: Foot of perpendicular = (2, 1). Image = (‑2, 3). - A point P(3, 7) is reflected in the line y = x + 2 to give point P'. Find P'.
Answer: Foot = (5, 7). Image = (7, 5). - State the rule for reflecting any point (x, y) in the line y = x.
Answer: Image is (y, x).