Ever wondered why a mirror image looks just right on graph paper?
Reflection in coordinate geometry is the math version of a mirror. It helps you flip points, lines or shapes over a line, just like a selfie in a bathroom mirror.
💡 In Simple Words: A reflection is a flip. Take any point on a graph, draw a line (the mirror), and the point’s twin appears on the opposite side, the same distance away.
What does “reflection” actually mean?
In geometry, a reflection is a transformation that produces a mirror image of a figure. The line you reflect across is called the line of reflection (think of it as the mirror). Every point and its image are equally far from that line, and the line joins them at a right angle.
How to find the image of a point – step by step
If you need to locate the reflected point, follow these four easy steps. The process works for any straight line, not just the axes.
Quick formulas for the most common mirrors
Memorising a few shortcuts saves time during the exam. Below is a handy table that tells you the image (x',y') of a point (x,y) after reflecting across popular lines.
| Line of reflection | Equation | Image of (x, y) |
|---|---|---|
| x‑axis | y = 0 | (x, -y) |
| y‑axis | x = 0 | (-x, y) |
| y = x | y - x = 0 | (y, x) |
| y = -x | y + x = 0 | (-y, -x) |
| General line ax + by + c = 0 | ax + by + c = 0 | Use perpendicular‑midpoint method (see steps) |
Worked example 1: reflecting a point across the x‑axis
Find the image of P(4, -3) when reflected in the x‑axis.
- Step 1: The mirror is the x‑axis, so y = 0.
- Step 2: The formula for the x‑axis is (x, -y). Plug in the numbers: (4, -(-3)) = (4, 3).
So the reflected point P' is (4, 3). Easy, right?
Worked example 2: reflecting a point across the line y = x
Find the image of Q(2, 7) when reflected in the line y = x.
- Step 1: The line y = x swaps the coordinates.
- Step 2: Apply the rule (x, y) → (y, x). So Q' becomes (7, 2).
Check: The midpoint of Q and Q' is ((2+7)/2, (7+2)/2) = (4.5, 4.5), which lies on y = x, confirming the reflection.
Worked example 3: reflecting across a slant line (general case)
Reflect R(5, 1) across the line 2x + y - 4 = 0.
- Write the line equation: 2x + y - 4 = 0.
- Find the perpendicular line through R. Its slope is the negative reciprocal of the original line’s slope. Original slope = -2 (because y = -2x + 4). Perpendicular slope = 1/2.
- Equation of perpendicular through R: y - 1 = (1/2)(x - 5).
- Solve the two equations together to get the intersection point M (the foot of the perpendicular). After a bit of algebra, M = (2, 0).
- Midpoint M is exactly halfway between R and its image R'. Use the midpoint formula: M = ((x_R + x_R')/2, (y_R + y_R')/2). Plug M(2,0) and R(5,1) to find R': x_R' = 2*2 - 5 = -1, y_R' = 2*0 - 1 = -1.
Thus the reflected point R' is (-1, -1).
Tips to avoid common mistakes
- Remember the line of reflection is the *mid‑point* line, not a random line that passes through the point.
- When the mirror is a coordinate axis, use the quick formulas – they’re faster and less error‑prone.
- Check your work by confirming the original point, its image, and the mirror line are all perpendicular at the midpoint.
- Write the equation of the mirror line in standard form (ax + by + c = 0) before tackling a slant line.
📝 Likely Exam Questions
- Question: Find the image of A(‑2, 5) when reflected in the y‑axis.
Answer: Using the y‑axis rule (‑x, y), the image is A'(2, 5). - Question: A point B(3, ‑4) is reflected in the line y = x. Write the coordinates of B'.
Answer: Swap the coordinates: B'(‑4, 3). - Question: Reflect C(1, 2) in the line 3x ‑ 4y + 12 = 0. Give the coordinates of C'.
Answer: Follow the perpendicular‑midpoint method. The foot of the perpendicular is M(2, 3). Using the midpoint formula, C' = (3, 4). - Question: State the image of the point D(‑6, 0) after reflection in the x‑axis and then in the line y = ‑x.
Answer: First reflection gives D1(‑6, 0) → (‑6, 0) (no change because y‑coordinate is zero). Second reflection swaps and changes signs: D2 = (0, 6). - Question: Explain why the distance from a point to its image is twice the distance from the point to the line of reflection.
Answer: The line of reflection is the midpoint of the segment joining the point and its image. Hence the segment is split into two equal halves, making the total distance twice the half‑distance.