Lesson 3.1: Geometric Transformations & Rigid Motions

Move Shapes Without Changing Shape

Translate, rotate, and reflect figures. Understand invariants: lengths, angle measures, and parallelism stay the same under rigid motions.

Learning Objectives

Shift by vector $\langle a,\ b\rangle$ : $(x,\ y)\to(x+a,\ y+b)$

About origin by 90° CW: $(x,\ y)\to(y,\ -x)$

Across x-axis: $(x,\ y)\to(x,\ -y)$

Angle $\theta$ about origin: $(x,\ y)\to(x\cos\theta-y\sin\theta,\ x\sin\theta+y\cos\theta)$

Across y-axis: $(x,\ y)\to(-x,\ y)$ ; Across $y=x$ : $(x,\ y)\to(y,\ x)$

Rigid motions preserve length, angle, area, and parallelism; rotations/reflections change orientation, translations keep it.

Rotate △ABC with A $(1,\ 2)$ , B $(3,\ 1)$ , C $(2,\ 4)$ 90° clockwise about origin.

Rule: $(x,\ y)\to(y,\ -x)$

A' $(2,\ -1)$ , B' $(1,\ -3)$ , C' $(4,\ -2)$

Reflect across x-axis then across y-axis. Show the composition equals a 180° rotation about origin.

Across x-axis: $(x,\ y)\to(x,\ -y)$ , then y-axis: $(x,\ -y)\to(-x,\ -y)$

Which equals rotation by $\pi$ : $(x,\ y)\to(-x,\ -y)$

Translate by $(-h,-k)$ , rotate by $\theta$ , then translate back by $(h,k)$ .

Use T_−P ∘ R_θ ∘ T_P.

Translations/rotations preserve orientation; reflections reverse orientation. Composition rule predicts final orientation.

1) Translate (−2, 5) by ⟨4, −3⟩, then reflect across x-axis

Solution

After translate → (2, 2); reflect x-axis → (2, −2)

2) Rotate (3, −1) CCW 90° about origin, then reflect across y=x

Solution

90° CCW: (x,y)→(−y,x) → (1,3); reflect y=x → (3,1)

3) Give a sequence of rigid motions mapping segment AB to A'B' with same length and direction

Hint

Translate A to A', then rotate about A' to align directions.