Congruence: Defining Rotations, Reflections, and Translations (CCSS.G-CO.4)
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Common Core High School Geometry › Congruence: Defining Rotations, Reflections, and Translations (CCSS.G-CO.4)
In triangle ABC, AB = 5, BC = 7, and ∠ABC = 60°. The triangle is rotated 90° about a point in the plane to form triangle A′B′C′.
After a 90° rotation, what is the measure of ∠A′B′C′?
120°
30°
60°
300°
Explanation
Rotations are rigid motions that preserve lengths and angle measures, so ∠A′B′C′ = ∠ABC = 60°. Distractors: 120° assumes the angle doubles after a 90° rotation (it does not), 30° is an arithmetic slip, and 300° reflects a misconception about orientation or reflex angles—interior angle measures of a triangle are preserved.
Which statement best defines a rotation in the plane?
A transformation determined by a center point and an angle that moves every point along a circular arc centered at that point, preserving distances and angle measures.
A transformation that slides every point the same distance along parallel lines in a fixed direction.
A transformation across a line that maps each point to a mirror image so the line is the perpendicular bisector of the segment joining them.
A transformation that changes lengths but preserves angle measures using a fixed scale factor.
Explanation
A rotation is specified by a center and an angle; each point travels along a circle centered at the rotation's center. It preserves segment lengths and angle measures. The other choices describe a translation, a reflection, and a dilation, respectively.
Which statement correctly defines a reflection across a line ℓ?
It moves each point along a circular path about a fixed center by a given angle.
It maps each point to a point on the opposite side of ℓ so that ℓ is the perpendicular bisector of the segment connecting the point and its image.
It slides every point the same distance in the same direction along parallel lines.
It shifts each point perpendicular to ℓ by a fixed distance, regardless of the point's location.
Explanation
In a reflection, the mirror line is the perpendicular bisector of the segment joining a point and its image. Reflections preserve lengths and angle measures but reverse orientation. The other choices misdescribe rotation, translation, or impose an incorrect uniform offset.
Which statement best characterizes a translation in terms of lines and segments?
It turns a figure about a fixed point by a specified angle.
It flips a figure across a line so that the line is the perpendicular bisector of each point-image segment.
It enlarges or reduces a figure from a center by a constant scale factor.
It moves every point the same distance in the same direction: for any point, the segment from the point to its image is congruent and parallel to the segment from any other point to its image, and lines map to parallel lines.
Explanation
A translation adds the same vector to all points: all point–image segments are congruent and parallel, and lines remain parallel. The other choices describe rotation, reflection, or dilation.
Which property is preserved under all rotations, reflections, and translations of a figure in the plane?
Lengths of segments and measures of angles.
Orientation (clockwise vs. counterclockwise order).
Parallel lines may become intersecting lines.
Perimeter and area change by the same nonzero scale factor.
Explanation
Rotations, reflections, and translations are rigid motions: they preserve distances, angle measures, collinearity, and parallelism. Reflections reverse orientation, so orientation is not preserved. Rigid motions do not turn parallel lines into intersecting ones, and they keep perimeter and area unchanged.