Questions: Geometric Transformations: Reflections

Question 1

Triangle ABC has vertices A(2, 3), B(5, 1), C(4, 6). It is reflected over the y-axis. What are the coordinates of the image vertex A'?

• A. (2, -3)
• B. (-2, 3)
• C. (-2, -3)
• D. (3, 2)

Answer: (B)

Reflection over the y-axis maps (x, y) to (−x, y): the x-coordinate changes sign, the y-coordinate stays the same. So A(2, 3) maps to A'(−2, 3). A common error is flipping both coordinates (which would be reflection over the origin, not the y-axis) or swapping them (which is reflection over y = x). Each axis of reflection has its own rule: y-axis changes x, x-axis changes y, and y = x swaps them.

Question 2

A figure is reflected over the line y = x. The original figure has vertices labeled clockwise as P, Q, R. After the reflection, how are the image vertices oriented?

• A. Clockwise — reflections preserve orientation
• B. Counterclockwise — reflections reverse orientation
• C. Clockwise — the line y = x is a special case that preserves orientation
• D. The orientation depends on the specific coordinates of P, Q, R

Answer: (B)

All reflections reverse orientation — this is a fundamental property that distinguishes reflections (indirect isometries) from translations and rotations (direct isometries). If vertices are labeled clockwise before a reflection, they are labeled counterclockwise after, for any line of reflection. The line y = x has no special exception. Think of it physically: a reflection is like lifting a figure off the table and flipping it over — the mirror image of a clockwise-labeled triangle is always counterclockwise.

Question 3

A reflection is a rigid motion, which means it preserves the distances between all points and the measures of all angles in the figure.

• T. True
• F. False

Answer: True

Correct. Reflections are isometries (rigid motions) — the image is always congruent to the original figure. Distance between any two points is preserved, and all angle measures are preserved. What a reflection changes is orientation: clockwise becomes counterclockwise. This distinguishes reflections from non-rigid transformations like dilations, which scale distances.

Question 4

The line of reflection is expected to be one of the coordinate axes or the line y = x — reflections over other lines are not standard geometric transformations.

• T. True
• F. False

Answer: False

A reflection can be defined over any line in the plane — horizontal, vertical, diagonal, or any oblique line like y = 2x + 1. The coordinate axis reflections and y = x are taught first because they have clean algebraic rules, but they are special cases of the general definition. The perpendicular bisector definition works for any line: the line of reflection is the perpendicular bisector of the segment connecting each original point to its image. Reflections over general lines require more computation (finding the foot of the perpendicular) but are fully standard transformations.

Question 5

Explain what it means for the line of reflection to be the 'perpendicular bisector' of each point and its image, and why this fully determines where every point maps to.

Model answer: For any point P and its image P' under a reflection, the line of reflection sits exactly halfway between them (bisects the segment PP') and crosses PP' at a right angle (is perpendicular to it). This two-part condition uniquely determines P': given P and the line of reflection, there is exactly one point P' such that the line perpendicularly bisects PP'. To find P', you drop a perpendicular from P to the line, find the foot F of that perpendicular, then extend the same distance on the other side: P' is the point such that F is the midpoint of PP' and PP' is perpendicular to the line.

The perpendicular bisector definition works for any line of reflection and explains why the coordinate rules work. For reflection over the x-axis: the midpoint of (x,y) and (x,−y) is (x,0) which lies on the x-axis, and the segment from (x,y) to (x,−y) is vertical (perpendicular to the horizontal x-axis). Both conditions of the perpendicular bisector are satisfied, confirming the rule is correct.