Regents Geometry · Math Enrichment
Regents Geometry Non-Rigid Transformations: Dilation, Stretch, and Compression
Dilations, stretches, and compressions — the non-rigid transformations that change size (and sometimes shape) on the NY Regents Geometry exam. Here is what they are, how the coordinate rules work, why area scales as k², and how SOMATH's math enrichment on the Upper West Side prepares students to master them.
The short answer: a non-rigid transformation changes the size of a figure on the coordinate plane. Because size changes, so does area. The three tested in Regents Geometry are dilation (resizes proportionally from a center point — the only non-rigid transformation that keeps shape), stretch (changes size in one direction), and compression (shrinks in one or both directions). Under a dilation with scale factor k, side lengths and perimeter multiply by |k|, but area multiplies by k². Only dilation produces similar figures; stretch and compression change shape.
This is Part 2 of our transformations pair. Part 1 covered rigid transformations — translation, reflection, and rotation. Together, the two posts give Upper West Side Regents Geometry students the full transformation toolkit the June and January Regents exams draw from.
The pair, in one sentence: rigid transformations (translation, reflection, rotation) move a figure without changing it → congruent images. Non-rigid transformations (dilation, stretch, compression) change size → non-congruent images. Dilation is the bridge — it changes size but keeps shape, so its images are similar to the original.
SOMATH — School of Math on the Upper West Side at 226 West 79th Street, first floor, phone (646) 668-6151 — covers dilations and similarity across our Young Fermats (grades 5–8) and high-school Geometry tracks. Every family starts with a free 60-minute in-person evaluation and a written diagnostic within 48 hours — yours to keep whether you enroll or not.
What is a non-rigid transformation?
A non-rigid transformation is any movement of a figure on the coordinate plane that changes the figure's size. Unlike rigid transformations (translation, reflection, rotation) — which preserve size, shape, side lengths, angles, perimeter, and area — non-rigid transformations do not preserve size, and typically do not preserve area, side lengths, or perimeter.
The three non-rigid transformations covered in Regents Geometry are:
- Dilation — resizes the figure proportionally from a center point. Keeps shape. Its images are similar to the original.
- Stretch — changes size in one direction only (horizontal or vertical). Changes shape.
- Compression — shrinks the figure in one or both directions. Changes shape.
Angles are preserved under all three of these when applied uniformly in one direction — but sides and area change. That is the whole point of "non-rigid": the figure is no longer congruent to itself after the transformation.
The three non-rigid transformations, in detail
Dilation — resize from a center point
A dilation resizes every point of the figure by a scale factor k, measured from a fixed center of dilation. On the Regents, the center is almost always the origin.
Coordinate rule (center at origin, scale factor k): (x, y) → (kx, ky)
Multiply every x and y by the same scale factor.
- If k > 1 → the image is an enlargement (bigger than the original).
- If 0 < k < 1 → the image is a reduction (smaller than the original).
- If k = 1 → the image is identical to the original (trivial case).
- If k is negative → the image is on the opposite side of the center of dilation (still resized by |k|).
Example (k = 2): dilate the rectangle with vertices (−3, 1), (−1, 1), (−1, 2), (−3, 2) by scale factor 2 about the origin. Apply (x, y) → (2x, 2y). New vertices: (−6, 2), (−2, 2), (−2, 4), (−6, 4). Same rectangle, twice as tall and twice as wide.
Effect on measurements:
- Side lengths are multiplied by |k|.
- Perimeter is multiplied by |k|.
- Area is multiplied by k².
- Angles remain the same.
- Shape is the same — the image is similar to the pre-image.
Stretch — change size in one direction
A stretch multiplies coordinates in one direction only. A horizontal stretch by factor k (with k > 1) makes the figure wider. A vertical stretch makes it taller.
Coordinate rules:
- Horizontal stretch: (x, y) → (kx, y) (k > 1)
- Vertical stretch: (x, y) → (x, ky) (k > 1)
Example (horizontal stretch, k = 2): stretch the rectangle with vertices (−2, 1), (−1, 1), (−1, 2), (−2, 2) horizontally by factor 2. Apply (x, y) → (2x, y). New vertices: (−4, 1), (−2, 1), (−2, 2), (−4, 2). The rectangle is now twice as wide but the same height — a different shape.
Effect on measurements:
- Side lengths change in one direction only (widths or heights).
- Perimeter changes.
- Area is multiplied by |k| (the scale factor in the stretched direction).
- Angles remain the same (no slant is introduced).
- Shape changes — the image is not similar to the pre-image.
Compression — shrink in one or both directions
A compression uses the same coordinate-rule format as a stretch, but with a scale factor k between 0 and 1. Instead of getting bigger, the figure shrinks in the compressed direction.
Coordinate rules (0 < k < 1):
- Horizontal compression: (x, y) → (kx, y)
- Vertical compression: (x, y) → (x, ky)
Example (horizontal compression, k = 1/2): compress the rectangle with vertices (−4, 1), (−2, 1), (−2, 3), (−4, 3) horizontally by factor 1/2. Apply (x, y) → (0.5x, y). New vertices: (−2, 1), (−1, 1), (−1, 3), (−2, 3). The rectangle is now half as wide.
Effect on measurements:
- Side lengths shrink by factor k in the compressed direction.
- Perimeter decreases.
- Area is multiplied by k (0 < k < 1), so it decreases.
- Angles remain the same.
- Shape changes — the image is not similar to the pre-image.
The summary table Regents students memorize
Every SOMATH Geometry student learns this table — it collapses the entire chapter into a single grid.
| Type | Changes size? | Changes shape? | Side lengths | Perimeter | Area | Example rule |
|---|---|---|---|---|---|---|
| Dilation | Yes | No (similar) | Multiplied by |k| | Multiplied by |k| | Multiplied by k² | (x, y) → (kx, ky) |
| Horizontal stretch | Yes | Yes (unless k = 1) | Width scaled by k | Changes | Multiplied by |k| | (x, y) → (kx, y) |
| Vertical stretch | Yes | Yes (unless k = 1) | Height scaled by k | Changes | Multiplied by |k| | (x, y) → (x, ky) |
| Horizontal compression | Yes | Yes (unless k = 1) | Width scaled by k (0<k<1) | Decreases | Multiplied by k (0<k<1) | (x, y) → (kx, y), 0<k<1 |
| Vertical compression | Yes | Yes (unless k = 1) | Height scaled by k (0<k<1) | Decreases | Multiplied by k (0<k<1) | (x, y) → (x, ky), 0<k<1 |
The k² rule for area under a dilation
This is the single most tested fact about non-rigid transformations on the Regents Geometry exam: under a dilation with scale factor k, area is multiplied by k². Not by k. By k-squared.
Why? Area is a two-dimensional measurement. When you scale both the width and the height by k, area — which is width times height — becomes k × k = k² times the original area.
Quick reference:
- k = 2 → area is multiplied by 4
- k = 3 → area is multiplied by 9
- k = 1/2 → area is multiplied by 1/4 (area is one-quarter of the original)
- k = 1/3 → area is multiplied by 1/9
Volume follows the same logic in three dimensions: under a dilation with scale factor k, volume is multiplied by k³. That fact appears on Regents Geometry Part III and Part IV problems involving 3D solids.
Worked example: a Regents-style dilation question
Problem. Triangle ABC has vertices A(1, 2), B(4, 2), C(4, 6). The image of triangle ABC under a dilation centered at the origin with scale factor k = 3 is triangle A'B'C'. (a) Find the coordinates of A', B', and C'. (b) State the ratio of the area of triangle A'B'C' to the area of triangle ABC.
Solution, part (a). Apply the dilation rule (x, y) → (3x, 3y):
- A(1, 2) → A'(3, 6)
- B(4, 2) → B'(12, 6)
- C(4, 6) → C'(12, 18)
Solution, part (b). Under a dilation with scale factor k, area is multiplied by k². Here k = 3, so area is multiplied by 3² = 9.
Answer: the ratio of the area of A'B'C' to the area of ABC is 9 to 1. Since the two triangles are related by a dilation, they are similar (but not congruent).
That is a 4-credit question on the Regents Geometry Part III — but a student who has the coordinate rule and the k² rule memorized answers it in under two minutes.
Rigid vs. non-rigid: the full comparison
| Property | Rigid transformations (translation, reflection, rotation) |
Non-rigid: dilation | Non-rigid: stretch / compression |
|---|---|---|---|
| Preserves size? | Yes | No | No |
| Preserves shape? | Yes | Yes (similar) | No (unless k = 1) |
| Produces congruent figures? | Yes | No | No |
| Produces similar figures? | Yes (special case) | Yes | No |
| Effect on area | Unchanged | Multiplied by k² | Multiplied by |k| (stretch direction only) |
| Effect on angles | Preserved | Preserved | Preserved |
This distinction shows up on the Regents in two big ways: (1) any question that says the word "congruent" is a rigid-transformations problem; (2) any question that says the word "similar" involves a dilation (usually combined with rigid motions). Read the wording carefully — that single word tells you which toolkit to use.
How SOMATH teaches non-rigid transformations (K–12 arc)
At SOMATH we introduce scaling and proportional reasoning gradually across our grade-level tracks, so by the time a student sees dilations on the Regents Geometry exam, the ideas are second nature.
- Little Newtons (grades K–2): comparing sizes, doubling and halving shapes made from paper. Intuition first, no coordinate plane yet.
- Kid Einsteins (grades 3–5): ratios, proportional reasoning, scale drawings, coordinate plane basics.
- Young Fermats (grades 5–8): dilations on the coordinate plane, ratio and proportion, similarity, why area scales as k² and volume as k³. Students derive these rules themselves before we hand them the summary.
- High-school Geometry: full Regents-level treatment. Coordinate rules memorized, similarity proofs, composing a dilation with rigid motions, and applying to 3D volume questions.
Our high-school Geometry teachers hold degrees from Harvard, Northwestern, Columbia, and NYU. If your child is preparing for the January or June Regents Geometry, transformations (both rigid and non-rigid) are among the highest-yield topics to build depth in. Read our rigid transformations companion post and our NYS Regents math prep guide for the full arc.
Quick memory tips for the Regents
- Non-rigid = size changes. The word "non-rigid" literally means the figure is no longer rigid — its size and often its shape change.
- Dilation preserves shape, everything else does not. If a Regents question refers to similar figures, it's about a dilation. Stretch and compression change shape and don't produce similar figures.
- Area scales as k² under a dilation, not as k. This is the trap the Regents sets most often. Sides go by k, area goes by k², volume goes by k³.
- Same coordinate rule for stretch and compression. The only difference is whether k > 1 (stretch) or 0 < k < 1 (compression). Look at the scale factor first.
- Similarity is rigid motions + dilation. To prove two figures are similar on the Regents, describe a sequence of rigid transformations followed by (or preceded by) a single dilation.
Book a free math enrichment evaluation
If your child is heading into Regents Geometry — or into the middle-school Common Core topics that build toward similarity and dilations — the best move is to see exactly where they are today. Book a free 60-minute in-person evaluation at SOMATH. Your child works one-on-one with a SOMATH teacher, and you receive a written diagnostic within 48 hours — specifically what your child has mastered, where the gaps are, and what to work on next. Yours to keep whether you enroll or not.
SOMATH is at 226 West 79th Street, first floor, between Broadway and Amsterdam. Phone (646) 668-6151. See our weekly class schedule or browse all courses grades 1–12.
Related reading: Regents Geometry Rigid Transformations: SOMATH's UWS Math Enrichment Guide · Elementary Math Enrichment on the Upper West Side (K–5) · After School Math Program on the Upper West Side (2026) · Best Math Enrichment on the Upper West Side.
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