Rotate figures about the origin

A rotation about the origin turns each point and figure through a specified angle while preserving lengths, angle measures, and overall shape; for quarter- and half-turns, coordinates are transformed by rules such as $(x,y)\rightarrow(-y,x)$ , $(x,y)\rightarrow(-x,-y)$ , and $(x,y)\rightarrow(y,-x)$ , depending on direction. Reasoning includes tracking signs and coordinate order and distinguishing clockwise from counterclockwise rotation; arbitrary-angle formulas and trigonometric methods are not included.

Detailed Explanation: Rotate figures about the origin

To rotate a figure about the origin, apply the rotation rule to every vertex. The origin is the point $(0,0)$ , and the figure keeps the same size and shape.

For a $90^\circ$ counterclockwise rotation, use:

(x,y)\rightarrow(-y,x)

This means:

Change the order of the coordinates.
Make the new first coordinate the opposite of the old $y$ -coordinate.
Keep the old $x$ -coordinate as the new $y$ -coordinate.

Example

Rotate triangle $ABC$ $90^\circ$ counterclockwise about the origin.

A(1,2),\qquad B(4,2),\qquad C(1,5)

Apply $(x,y)\rightarrow(-y,x)$ to each point.

For $A(1,2)$ :

(1,2)\rightarrow(-2,1)

So, $A'(-2,1)$ .

For $B(4,2)$ :

(4,2)\rightarrow(-2,4)

So, $B'(-2,4)$ .

For $C(1,5)$ :

(1,5)\rightarrow(-5,1)

So, $C'(-5,1)$ .

The rotated triangle has vertices:

\boxed{A'(-2,1),\ B'(-2,4),\ C'(-5,1)}

Remember the common quarter- and half-turn rules:

$90^\circ$ counterclockwise: $(x,y)\rightarrow(-y,x)$
$180^\circ$ : $(x,y)\rightarrow(-x,-y)$
$90^\circ$ clockwise: $(x,y)\rightarrow(y,-x)$

Learn by doing: Rotate figures about the origin

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Cartesian Grid - Rotation of Point (Grid to Coordinates) around Origin

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