Rotate figures about the origin

A rotation about the origin turns each point through a specified angle while preserving distances, angle measures, and orientation; for quarter-turns and a half-turn, coordinates follow the rules $(x,y)\to(-y,x)$ , $(x,y)\to(-x,-y)$ , and $(x,y)\to(y,-x)$ for $90^\circ$ , $180^\circ$ , and $270^\circ$ counterclockwise, respectively. The understanding includes distinguishing clockwise from counterclockwise direction and applying the rule consistently to figures; arbitrary-angle rotations, trigonometric formulas, and matrix representations are beyond this scope.

Detailed Explanation: Rotate figures about the origin

A rotation about the origin moves every point the same amount around $(0,0)$ . The figure keeps the same size and shape.

For quarter-turns, use these coordinate rules:

$90^\circ$ counterclockwise: $(x,y)\to(-y,x)$
$180^\circ$ : $(x,y)\to(-x,-y)$
$270^\circ$ counterclockwise: $(x,y)\to(y,-x)$

A $90^\circ$ clockwise turn is the same as a $270^\circ$ counterclockwise turn.

Example

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

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

Because the rotation is $90^\circ$ counterclockwise, use

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

Apply the rule to each vertex:

For $A(1,2)$ :

(1,2)\to(-2,1)

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

For $B(4,2)$ :

(4,2)\to(-2,4)

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

For $C(1,5)$ :

(1,5)\to(-5,1)

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

The rotated triangle has vertices

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

To rotate a whole figure, apply the same coordinate rule to every vertex, then connect the new points in the same order.

Learn by doing: Rotate figures about the origin

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Cartesian Grid - Rotation of Shape around Origin

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