Describe transformations using coordinates

A learner describes translations, reflections across the coordinate axes, and rotations about the origin by 90°, 180°, or 270° by stating how each point’s $(x,y)$ coordinates change, and uses these rules to determine the image of a figure. This understanding connects coordinate relationships with geometric transformations that preserve lengths and angle measures; it does not extend to arbitrary centers or angles, matrix representations, or general transformation formulas.

Detailed Explanation: Describe transformations using coordinates

A transformation changes every point in a figure using the same coordinate rule. Apply the rule to each vertex, then connect the new points in the same order.

Coordinate rules

Translation right $a$ units and up $b$ units:

(x,y)\rightarrow (x+a,y+b)

Reflection across the $x$ -axis:

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

Reflection across the $y$ -axis:

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

Rotation $90^\circ$ counterclockwise about the origin:

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

Rotation $180^\circ$ about the origin:

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

Rotation $270^\circ$ counterclockwise about the origin:

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

For a rotation, always switch the positions of $x$ and $y$ as shown. Then check whether either coordinate changes sign.

Worked example

Triangle $ABC$ has vertices

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

Rotate the triangle $90^\circ$ counterclockwise about the origin. Find the coordinates of the image.

Use the rule

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

Apply it to each vertex:

\begin{aligned} A(1,2)&\rightarrow A'(-2,1),\\ B(4,2)&\rightarrow B'(-2,4),\\ C(1,5)&\rightarrow C'(-5,1). \end{aligned}

Therefore, the rotated triangle has vertices

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

The transformation is described by saying: “Each point is rotated $90^\circ$ counterclockwise about the origin, using the coordinate rule $(x,y)\rightarrow(-y,x)$ .”

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

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