12 multiple-choice questions, progressively harder.
Which instruction correctly follows a composition of two motions?
Solution
Correct answer: B
The first motion supplies an intermediate image. The second motion takes that intermediate image as its starting point. This follows one continuous ordered sequence.
First translate K(−1,5)K(-1,5)K(−1,5) one unit right and three down, then reflect across the xxx-axis. What is K′′K''K′′?
Correct answer: D
The translation gives K′(0,2)K'(0,2)K′(0,2). Reflection across the horizontal axis negates the new height.
−2=−(2)-2=-(2)−2=−(2)
Thus K′′(0,−2)K''(0,-2)K′′(0,−2).
What can happen when the order of two rigid motions is reversed?
Correct answer: A
Some orders give different destinations, such as a slide across a mirror before or after reflecting. Other pairs give the same mapping in either order, including perpendicular mirror reflections. Each sequence still consists of rigid motions.
What is the inverse of a translation with displacement pair (−7,2)(-7,2)(−7,2)?
Correct answer: C
Undo each horizontal and vertical change with its opposite.
−7+7=0-7+7=0−7+7=0
2−2=02-2=02−2=0
The inverse moves seven right and two down.
Which motion undoes reflection across line m?
Reflecting again across the same mirror returns every off-mirror image to its original partner. Points on the mirror stay fixed in both steps.
Which motion undoes a 68-degree counterclockwise rotation about C?
Keep the center and reverse the directed turn. Counterclockwise is positive.
68∘−68∘=0∘68^\circ-68^\circ=0^\circ68∘−68∘=0∘
Every point returns along its circle to its original position.
Reflect first in the right-hand mirror, then in the parallel mirror six units to its left. Which single motion results?
Parallel mirrors give twice the perpendicular gap.
2(6)=122(6)=122(6)=12
The direction is from the first mirror toward the second, which is left.
The directed angle from the first mirror to the second is 22 degrees clockwise. Their intersection is O. What is the composition of the two reflections in that order?
Double the directed mirror angle.
2(22∘)=44∘2(22^\circ)=44^\circ2(22∘)=44∘
The order gives clockwise direction. The intersection stays fixed and is the rotation center.
Which description meets the definition of a glide reflection?
The translation must run along the mirror and have nonzero displacement. The reflection and this parallel translation commute. A zero translation leaves an ordinary reflection.
What establishes that two whole figures are congruent in this course?
Congruence is defined by an exact whole-figure match using translations, reflections, and rotations. Selected equal measurements or an arbitrary mapping do not supply such a sequence.
A rigid sequence sends AAA to LLL, BBB to MMM, and CCC to NNN. Which side corresponds to BCBCBC?
Follow the endpoints separately. B matches M and C matches N, so the image of side BC is side MN.
The map (x,y)↦(3x,y)(x,y)\mapsto(3x,y)(x,y)↦(3x,y) sends (0,0)(0,0)(0,0) and (1,0)(1,0)(1,0) to points three units apart. What does this show?
The original horizontal distance is 1. The new horizontal distance is 3.
1≠31\ne31=3
A rigid mapping preserves every distance, so this one fails the requirement.
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