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Rigid motions

Reflections, rotations, and translations are all rigid motions — they preserve every distance and every angle. Useful coordinate rules:

Reflection over the x-axis: $(x,y) \to (x,-y)$
Reflection over the y-axis: $(x,y) \to (-x,y)$
Reflection over the line $y=x$ : $(x,y) \to (y,x)$
90° rotation about the origin (counterclockwise): $(x,y) \to (-y,x)$
Translation by vector $(a,b)$ : $(x,y) \to (x+a, y+b)$

Homothety

A homothety (scaling transformation) centered at a point with ratio $k$ multiplies every distance from that center by $k$ , while preserving angles and directions — it's the transformation underlying similar figures. Coordinates scale as:

(x,y) \to \big(x_0 + k(x-x_0),\ y_0+k(y-y_0)\big) \quad\text{(center } (x_0,y_0)\text{)}

Unlike the rigid motions above, homothety changes distances (unless $k=\pm1$ ) — but it never changes angles, which is why it always produces a similar copy of the original figure.

Worked example

Problem: Reflect the point $(3, 5)$ across the line $y=x$ , then rotate the result $90°$ counterclockwise about the origin. Find the final point.

Solution: Reflecting across $y=x$ swaps coordinates: $(3,5) \to (5,3)$ . Rotating $90°$ counterclockwise sends $(x,y) \to (-y,x)$ , so $(5,3) \to (-3,5)$ .

Practice problems

1.Reflect $(2,-7)$ across the x-axis.

2.Rotate $(4,0)$ by $90°$ counterclockwise about the origin.

3.A homothety centered at the origin with ratio $3$ sends a triangle of area 5 to a new triangle. Find the new area.

4.Translate the point $(-2,3)$ by the vector $(5,-1)$ .

Transformations

Rigid motions

Homothety

Practice problems