6.1: Rigid Motions | Geometry Honors | Code Scholars Tutoring

How it works

Translations, rotations, and reflections preserve distances and angle measures. They produce congruent figures. Coordinate rules must specify the center or axis; a standard origin rule does not apply to an arbitrary center.

R_{90^\circ}(x,y)=(-y,x),\quad F_{y\text{-axis}}(x,y)=(-x,y)

Worked examples

Example 1

Rotate $\displaystyle (3,1)$ by $\displaystyle 90^\circ$ counterclockwise about the origin.

Apply $\displaystyle (x,y)\mapsto(-y,x)$ to obtain $\displaystyle (-1,3)$ .

Example 2

Reflect $\displaystyle (-2,5)$ across the $\displaystyle x$ -axis.

The horizontal coordinate stays fixed and the vertical coordinate changes sign: $\displaystyle (-2,-5)$ .

Problems

Problem 6.1.1

Translate $\displaystyle A=(0,0)$ , $\displaystyle B=(4,0)$ , $\displaystyle C=(0,3)$ by $\displaystyle (2,1)$ .

Triangle with vertices 0,0; 4,0; 0,3. The solution shows its translate by 2,1. — Add the translation vector to every vertex.

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The image vertices are $\displaystyle A\prime=(2,1)$ , $\displaystyle B\prime=(6,1)$ , and $\displaystyle C\prime=(2,4)$ . All side lengths are preserved.

Triangle with vertices 0,0; 4,0; 0,3. The solution shows its translate by 2,1. The solution adds labeled relationships. — The image vertices are (2,1), (6,1), and (2,4).

Problem 6.1.2

Rotate $\displaystyle (2,-4)$ by $\displaystyle 180^\circ$ about the origin.

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Negate both coordinates: $\displaystyle (-2,4)$ .

Problem 6.1.3

Reflect $\displaystyle (4,1)$ across $\displaystyle y=x$ .

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Interchange coordinates to obtain $\displaystyle (1,4)$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 6.1.1

Problem 6.1.2

Problem 6.1.3

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