Orienting Yourself: The Use of Coordinates
Reflection keeps lengths
See what happens to a point's coordinates when you reflect it in an axis.
What if some coordinates are negative? In Fig. 1.9, triangle ADM is reflected in the y-axis, as if a mirror stood on the y-axis.
Reflecting in the y-axis flips left and right: x changes sign, y stays.
- A (3, 4) → A' (–3, 4)
- M (9, 6) → M' (–9, 6)
- D (7, 1) → D' (–7, 1)
Work out the sides of the image:
- C'D' = –3 – (–7) = 4 and A'C' = 4 – 1 = 3, so A'D' = √(4² + 3²) = 5;
- D'M' = √((–2)² + 5²) = √29;
- M'A' = √((–6)² + 2²) = √40.
These are exactly the lengths of the original triangle. A reflection keeps the side lengths, so the shape and size stay the same too. What changed is where the triangle is, and it now faces the other way, like your face in a mirror.
Reflecting in the x-axis flips top and bottom instead: (x, y) → (x, –y). Lengths are kept again.
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