Regents Geometry
Regents Geometry: Which Transformation Changes a Rectangle's Area?
A NYS Regents Geometry multiple-choice question, worked line by line — plus the rigid-motion vs. dilation theory every Upper West Side high-school student needs before test day.
The Question (NYS Regents Geometry)
Which transformation would result in the area of a rectangle's image being different from the area of its pre-image?
- a reflection over the y-axis
- a translation 4 units to the right
- a rotation of 90° counterclockwise about the origin
- a vertical stretch of scale factor 3 with respect to y = 0
Answer: (4) a vertical stretch of scale factor 3 with respect to y = 0.
Options (1), (2), and (3) are rigid motions and preserve area. Option (4) is a non-rigid transformation that multiplies the vertical dimension by 3, so the image area is three times the pre-image area.
The Theory: Rigid Motions vs. Non-Rigid Transformations
Every transformation on the Regents Geometry exam falls into one of two families. Knowing which family a transformation belongs to answers area, congruence, and similarity questions almost instantly.
Rigid Motions (Isometries)
A rigid motion — also called an isometry — is a transformation that preserves the distance between every pair of points. There are exactly three:
- Translations — slide every point the same vector.
- Reflections — flip every point across a line.
- Rotations — turn every point through a fixed angle around a center.
Because rigid motions preserve distance, they also preserve length, angle measure, perimeter, and area. The pre-image and image are always congruent.
Composition matters: any composition of rigid motions is still a rigid motion, so glide reflections and double reflections also preserve area.
Non-Rigid Transformations
A non-rigid transformation changes distance between at least some pairs of points. On the Regents, the two you must know are:
- Dilations (scale factor k in every direction from a center) — multiply every length by k, so area is multiplied by k2.
- Stretches (scale factor k in one direction only) — multiply one dimension by k, so area is multiplied by k.
Non-rigid transformations produce a figure that is similar but not congruent to the original (or, for shears, neither).
Testing Each Option, Step by Step
Take any rectangle — say the corners A(0, 0), B(4, 0), C(4, 2), D(0, 2). Original area = 4 × 2 = 8 square units. Now check each choice.
(1) Reflection over the y-axis
The rule is (x, y) → (−x, y). The image corners become A′(0, 0), B′(−4, 0), C′(−4, 2), D′(0, 2). Sides are still 4 and 2. Area = 8. Rigid motion → area preserved.
(2) Translation 4 units to the right
The rule is (x, y) → (x + 4, y). The rectangle slides but does not deform. Area = 8. Rigid motion → area preserved.
(3) Rotation of 90° counterclockwise about the origin
The rule is (x, y) → (−y, x). The rectangle turns but its side lengths do not change. Area = 8. Rigid motion → area preserved.
(4) Vertical stretch of scale factor 3 with respect to y = 0
The rule is (x, y) → (x, 3y). Horizontal lengths are unchanged, but vertical lengths triple. New corners: A′(0, 0), B′(4, 0), C′(4, 6), D′(0, 6). New area = 4 × 6 = 24 square units — exactly 3 times the original.
Only option (4) changes the area. The correct choice is (4).
The 5-Second Shortcut for Test Day
On any Regents Geometry area-of-image question, scan the four options and mentally sort each into "rigid" or "non-rigid":
- Words like translate, reflect, rotate, glide → rigid → area unchanged.
- Words like dilate, stretch, compress, scale factor → non-rigid → area changes.
The option that changes area is almost always the one non-rigid transformation in the list. This question is a textbook example.
Common Regents Mistakes on This Question Type
- Confusing rotation with dilation. A rotation turns the figure — it does not resize it. Only dilations and stretches resize.
- Forgetting that a k-factor dilation multiplies area by k2, not k. If the problem said "dilation of scale factor 3," the new area would be 9× the original — not 3×. Watch the wording carefully.
- Assuming "with respect to y = 0" means rotation. "With respect to a line" for a stretch means that line stays fixed and everything else moves perpendicular to it. The x-axis (y = 0) is fixed here; every other point moves vertically.
- Reading "stretch" as "translate." A stretch is a scaling in one direction, not a slide. Different word, different family.
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