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USA - Geometry Transformations, congruence, and proof

Translations, reflections, rotations

20 practice questions 2 video lessons Theory + worked examples

Translations, Reflections, and Rotations

California Geometry • Standard G-CO.5 • Transformations, Congruence & Proof

Translations, Reflections, and Rotations is a topic in Transformations, Congruence & Proof in the California Common Core State Standards. It is aligned to Standard G-CO.5, which requires students to develop definitions of rotations, reflections, and translations and to draw the image of a figure under a given rigid motion.

Translations, reflections, and rotations are rigid motions that slide, flip, or turn a figure while preserving its size and shape.

California Geometry › Transformations, Congruence & Proof › Translations, Reflections, and Rotations  —  Standard G-CO.5

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Theory

A rigid motion (or isometry) moves a figure without changing its size or shape — distances and angles are preserved. There are three:

  • Translation — slides every point the same distance and direction.
  • Reflection — flips the figure over a line (the line of reflection).
  • Rotation — turns the figure about a fixed point by an angle.
Rigid motions preserve congruence: the image is always congruent to the original.
Translation slides a figure A translation slides every point of a triangle the same distance and direction, here 3 units right. ΔABC image
A translation slides the triangle 3 units right.
Reflection flips a figure over a line A reflection over the y-axis flips the figure, sending each point (x, y) to (negative x, y). reflect over y-axis
A reflection over the \(y\)-axis flips the figure.

Common coordinate rules about the origin:

\[\text{translate: }(x,y)\to(x+a,\ y+b)\]
\[\text{reflect: } x\text{-axis }(x,-y),\ \ y\text{-axis }(-x,y),\ \ y=x\ (y,x)\]
\[\text{rotate: } 90^\circ(-y,x),\ \ 180^\circ(-x,-y),\ \ 270^\circ(y,-x)\]
translation adds to coordinates; reflections and rotations follow standard coordinate rules
Rotations here are counterclockwise about the origin, the positive direction.

How to apply a transformation

  1. Identify the type and its rule.
  2. Apply the rule to each vertex's coordinates.
  3. Plot the image points and connect them.
  4. Check the image is congruent to the original.
Example 1 — Translate a point
Translate \(A(2,-1)\) by \((x,y)\to(x+4,\ y+3)\).
Solution

Add the shift to each coordinate.

\(A'\)\(=\)\((2+4,\ -1+3)\)
\(=\)\((6,2)\)
the image is 6 comma 2
Example 2 — Reflect over the x-axis
Reflect \(B(3,5)\) over the \(x\)-axis.
Solution

Reflection over the \(x\)-axis negates the \(y\)-coordinate: \((x,y)\to(x,-y)\).

\(B'\)\(=\)\((3,-5)\)
the image is 3 comma negative 5
Example 3 — Rotate 90° about the origin
Rotate \(C(4,2)\) by \(90^\circ\) counterclockwise about the origin.
Solution

A \(90^\circ\) rotation uses \((x,y)\to(-y,x)\).

\(C'\)\(=\)\((-2,4)\)
the image is negative 2 comma 4
Example 4 — Identify the transformation
\(D(1,2)\to D'(-1,-2)\). What transformation is this?
Solution

Both coordinates change sign, \((x,y)\to(-x,-y)\), which is a rotation of \(180^\circ\) about the origin.

it is a 180-degree rotation about the origin

Common pitfalls

Reflection rules depend on the line. Over the \(x\)-axis negate \(y\); over the \(y\)-axis negate \(x\).
\(90^\circ\) rotation is \((-y,x)\), not \((y,-x)\). Watch the direction — counterclockwise is positive.
Rigid motions never resize. A dilation changes size and is not a rigid motion.

Frequently asked questions

What is a rigid motion?

A transformation that preserves distance and angle — a translation, reflection, or rotation. The image is congruent to the original.

What is the rule for reflecting over the x-axis?

\((x,y)\to(x,-y)\): keep \(x\), negate \(y\).

What is the rule for a 90-degree rotation about the origin?

Counterclockwise, \((x,y)\to(-y,x)\).

Do transformations change the size of a figure?

Rigid motions (translations, reflections, rotations) do not. Only a dilation changes size.

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