Generalizing Orientation and Congruence in Transformations on a Coordinate Plane(TEKS.Math.8.10.A)
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Texas 8th Grade Math › Generalizing Orientation and Congruence in Transformations on a Coordinate Plane(TEKS.Math.8.10.A)
Consider these examples: a hexagon is rotated $90^\circ$ clockwise; a triangle is slid 4 units right; a square is reflected across the $x$-axis; and a star is dilated by a factor of 2. Which transformations preserve orientation?
Reflections and translations
Rotations and translations
Reflections and dilations
Reflections only
Explanation
Orientation means the order/direction of the vertices (clockwise vs. counterclockwise) stays the same. Rotations and translations keep the figure facing the same way, so they preserve orientation (and congruence). Reflections flip the figure, reversing orientation (though they keep congruence). Dilations keep orientation but change size, so they do not preserve congruence unless the scale factor is 1.
Which transformations preserve congruence (same size and shape)? For example, think about a triangle rotated $180^\circ$, a square slid left, a pentagon reflected across the $y$-axis, and a hexagon dilated by factor 2.
Dilations only
Reflections only
Rotations and dilations
Rotations, translations, and reflections
Explanation
Congruence means same size and shape. The rigid motions—rotations, translations, and reflections—preserve congruence. Dilations change size (unless the scale factor is 1), so they do not preserve congruence.
Which option shows a transformation that preserves both orientation and congruence?
A triangle is rotated $180^\circ$ about the origin.
A parallelogram is reflected across the $y$-axis.
A square is dilated by scale factor $\tfrac{1}{2}$.
A pentagon is dilated by factor 3.
Explanation
Rotations and translations preserve both orientation (the figure's facing direction) and congruence (same size and shape). Reflections preserve congruence but reverse orientation. Dilations preserve orientation but change size, so they do not preserve congruence unless the scale factor is 1.
Which transformation reverses orientation but keeps the figure congruent?
A figure is translated 6 units down.
A figure is rotated $270^\circ$ counterclockwise.
A figure is reflected across the $x$-axis.
A figure is dilated by a factor of 3.
Explanation
Reflections flip the figure, reversing orientation (clockwise vs. counterclockwise order of vertices), but they are rigid motions, so they preserve congruence. Rotations and translations preserve both orientation and congruence. Dilations preserve orientation but change size, so they do not preserve congruence unless the scale factor is 1.
Which transformations do not preserve congruence?
Rotations and translations
Dilations with scale factor not equal to 1
Rotations, translations, and reflections
Reflections only
Explanation
Dilations change size, so they do not preserve congruence unless the scale factor is 1. Rotations and translations preserve both orientation and congruence. Reflections preserve congruence but reverse orientation.