Geometry · Lesson 10

Transformations & Intro to Trigonometry

Transformations move or resize geometric figures. Trigonometry connects angle measures to side ratios in right triangles — the foundation for all of precalculus and beyond.

Rigid Transformations

Rigid transformations preserve shape and size (congruence):

Translation: Slide. Every point moves the same vector (a, b). Rule: (x, y) → (x+a, y+b).
Rotation: Turn about a center point by an angle. 90° CCW: (x,y)→(−y,x).
Reflection: Flip over a line. Over x-axis: (x,y)→(x,−y). Over y-axis: (x,y)→(−x,y).

Non-Rigid Transformations

Dilation: Scale by factor k from a center. Rule: (x,y)→(kx,ky) from origin. If k>1: enlargement; 0<k<1: reduction. Produces similar (not congruent) figures.

Trigonometric Ratios (SOH-CAH-TOA)

In a right triangle with angle θ:

sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent

Memory trick: SOH-CAH-TOA

Special Angles

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3

Key Connection: Transformations map figures to congruent or similar images. The composition of two reflections over parallel lines equals a translation; over intersecting lines equals a rotation. These compositions connect all rigid motions.

Quick Check

1. A translation moves every point (x, y) by (+3, −2). Where does (1, 5) go?

(−2, 7)

(4, 3)

(3, 10)

(1, 3)

2. In a right triangle, sin θ equals:

Adjacent / Hypotenuse

Opposite / Hypotenuse

Opposite / Adjacent

Hypotenuse / Adjacent

3. A dilation with scale factor k = 2 centered at the origin maps (3, 4) to:

(3+2, 4+2)

(6, 8)

(1.5, 2)

(5, 6)

4. Which angle has tan θ = 1?

30°

45°

60°

90°

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