Congruence: Properties of Dilations and Scale Factor (CCSS.G-CO.13)
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Common Core High School Geometry › Congruence: Properties of Dilations and Scale Factor (CCSS.G-CO.13)
Triangle ABC is mapped to triangle A′B′C′ by the sequence of transformations: first translate by $(x,y)\to(x+2,,y-3)$, then rotate $90^\circ$ counterclockwise about the origin. Which statement correctly describes △ABC and △A′B′C′?
△ABC and △A′B′C′ are similar but not congruent because a rotation changes orientation.
△ABC and △A′B′C′ are congruent because a translation followed by a $90^\circ$ rotation are rigid motions that preserve side lengths and angle measures.
△ABC and △A′B′C′ are not congruent because the rule $(x,y)\to(x+2,,y-3)$ changes all side lengths by 5 units.
△ABC and △A′B′C′ are not congruent because rotations do not preserve angle measures.
Explanation
Translations and rotations are rigid motions; they preserve lengths and angles. Therefore △ABC and △A′B′C′ are congruent. The distractors incorrectly claim a change in size or loss of angle preservation.
Triangle ABC is mapped to triangle A′B′C′ by the sequence: reflect across the $y$-axis $(x,y)\to(-x,,y)$, then translate $(x,y)\to(x-3,,y+1)$, then rotate $180^\circ$ about the origin. Which statement correctly describes △ABC and △A′B′C′?
△ABC and △A′B′C′ are similar only because reflections reverse orientation and therefore change size.
△ABC and △A′B′C′ are not congruent because the translation $(x,y)\to(x-3,,y+1)$ changes side lengths by 3 units.
△ABC and △A′B′C′ are congruent because reflections, translations, and $180^\circ$ rotations are rigid motions that preserve lengths and angles.
△ABC and △A′B′C′ are not congruent because a $180^\circ$ rotation is equivalent to a dilation by factor $-1$.
Explanation
Reflections, translations, and $180^\circ$ rotations are all rigid motions, so they preserve side lengths and angle measures; the triangles are congruent. The distractors mistakenly treat rigid motions as dilations or claim length changes.
△ABC is mapped to △A′B′C′ by the translation $T: (x,y) \to (x+2, y-3)$ followed by a $90^\circ$ counterclockwise rotation about the origin. Which statement correctly describes △ABC and △A′B′C′?
△ABC and △A′B′C′ are similar but not congruent because the translation changes side lengths.
△ABC and △A′B′C′ are not congruent because a $90^\circ$ rotation changes angle measures.
△ABC and △A′B′C′ are congruent because a translation followed by a rotation are rigid motions that preserve lengths and angles.
△ABC and △A′B′C′ are congruent because a dilation with scale factor 1 preserves side lengths.
Explanation
Translations and rotations are rigid motions, so they preserve all distances and angle measures; hence the triangles are congruent. The distractors incorrectly claim length/angle changes or invoke dilation, which is not part of the mapping.
△ABC is mapped to △A′B′C′ by a reflection across the $y$-axis, then the translation $T: (x,y) \to (x-5, y+1)$. Which statement correctly describes △ABC and △A′B′C′?
△ABC and △A′B′C′ are not congruent because a reflection reverses orientation, which changes side lengths.
△ABC and △A′B′C′ are congruent because a reflection and a translation are rigid motions that preserve all distances and angle measures.
△ABC and △A′B′C′ are similar but not congruent because translating by $(-5,1)$ changes the perimeter.
△ABC and △A′B′C′ are not congruent because reflections preserve angles but not side lengths.
Explanation
Reflections and translations are rigid motions; they preserve lengths and angles (orientation may change under reflection, but congruence does not depend on orientation). The other options incorrectly assert changes in length or perimeter.