Ever tried to see your selfie in a mirror on a graph paper? That's exactly what reflection in coordinate geometry does – it creates a perfect copy on the other side of an invisible line.
💡 In Simple Words: Reflection flips a point or a shape over a line, just like a mirror flips your face. The original stays where it is, and the image shows up the same distance on the other side.
What is Reflection in Coordinate Geometry?
In plain language, a reflection is a transformation that produces a mirror image of a figure across a line called the line of reflection. The line works like a ruler that tells you how far to move each point.
Common Lines of Reflection and Their Formulas
Across the x‑axis
Imagine a horizontal mirror lying on the x‑axis. Every point (x, y) jumps straight down or up, keeping the x‑coordinate the same but flipping the sign of y. The rule is:
(x, y) → (x, ‑y)
Across the y‑axis
Now picture a vertical mirror on the y‑axis. The y‑coordinate stays, but x flips sign:
(x, y) → (‑x, y)
Across the line y = x
This line runs diagonally like a staircase. Swapping the coordinates does the trick:
(x, y) → (y, x)
Across the line y = ‑x
Another diagonal, but sloping the other way. Both coordinates swap and change sign:
(x, y) → (‑y, ‑x)
Across a vertical line x = k
k is a constant (like 2 or ‑3). The image is as far from the line as the original, but on the opposite side:
(x, y) → (2k ‑ x, y)
Across a horizontal line y = k
Similar idea, just vertical:
(x, y) → (x, 2k ‑ y)
Quick Reference Table
| Line of Reflection | Formula |
|---|---|
| x‑axis (y = 0) | (x, y) → (x, -y) |
| y‑axis (x = 0) | (x, y) → (-x, y) |
| y = x | (x, y) → (y, x) |
| y = -x | (x, y) → (-y, -x) |
| x = k | (x, y) → (2k - x, y) |
| y = k | (x, y) → (x, 2k - y) |
How to Reflect a Point – Step‑by‑Step
When the line isn’t one of the easy cases, follow these four moves:
Let’s walk through an example.
Example 1: Reflect (3, 4) across the line x = 1
1️⃣ The line is vertical at x = 1. 2️⃣ Distance from the point to the line is |3‑1| = 2 units to the right. 3️⃣ Move 2 units left of the line, landing at x = -1. 4️⃣ y‑coordinate doesn’t change. So the image is (-1, 4).
Example 2: Reflect (‑2, 5) across the line y = -2
Vertical distance = |5‑(‑2)| = 7. Go 7 units below – new y = -9. x stays ‑2. Image: (‑2, ‑9).
Reflecting a Line (or a Shape)
To reflect a whole line, reflect any two points on it and draw the new line through the images. The slope may change, but the distance between the original line and its mirror stays constant.
Example 3: Reflect the line 2x + y = 5 across the x‑axis
Pick two easy points: when x = 0, y = 5 → (0, 5). When y = 0, x = 2.5 → (2.5, 0). Reflect each across the x‑axis: (0, 5) → (0, ‑5) and (2.5, 0) → (2.5, 0) (unchanged). Now find the equation through (0, ‑5) and (2.5, 0). Slope = (0‑(‑5)) / (2.5‑0) = 5/2.5 = 2. Using point‑slope: y + 5 = 2(x‑0) → y = 2x‑5. So the reflected line is y = 2x ‑ 5.
Quick Checklist Before the Exam
- Identify the line of reflection correctly – is it a axis, a diagonal, or x = k / y = k?
- Remember the sign rules: flipping across a horizontal line changes the y‑sign, vertical changes the x‑sign.
- For x = k or y = k, use the “2k ‑ coordinate” formula.
- When in doubt, draw a quick sketch – the picture often tells you the answer.
- Check your answer by measuring the distance from the original point to the line and confirming the image is the same distance on the other side.
📝 Likely Exam Questions
- Find the image of the point (‑3, 2) when reflected in the line y = x.
Swap the coordinates → (2, ‑3). - Reflect the point (5, ‑1) across the line x = 2 and write the coordinates of the image.
Distance from line = |5‑2| = 3. New x = 2‑3 = ‑1. y stays ‑1 → (‑1, ‑1). - The line 3x – 4y = 12 is reflected in the x‑axis. Find the equation of the reflected line.
Replace y by –y: 3x – 4(‑y) = 12 → 3x + 4y = 12. - Given a triangle with vertices A(1,2), B(4,2), C(1,5), draw its image after reflecting in the line y = 2.
Only y‑coordinates change: new points A'(1,2), B'(4,2), C'(1,‑1). The triangle collapses to a line segment AB because A and B lie on the mirror line. - State the rule for reflecting any point across the line y = –x.
The rule is (x, y) → (‑y, ‑x).