Multiply every coordinate by the scale factor — the one transformation that changes size.
The Mnemonic
Multiply Every Coordinate by k
A dilation centered at the origin with scale factor k follows the rule: (x, y) → (kx, ky). Every coordinate gets multiplied by the same value k.
The value of k determines what kind of dilation occurs: k > 1 enlarges the shape. 0 < k < 1 reduces the shape (shrinks it, since multiplying by a fraction less than 1 makes values smaller). k = 1 leaves the shape completely unchanged. A NEGATIVE k both resizes AND rotates the shape 180° at the same time.
Every vertex moves directly away from (or toward) the center, scaled by exactly k
💡 Memory Trick
"Multiply every coordinate by the scale factor." (x,y) → (kx, ky) — the simplest of all four transformation rules, since both coordinates get the exact same treatment (multiply by k), unlike reflections and rotations where different coordinates get different rules.
Why It Works
Connecting Back to Similar Figures
This rule is the coordinate-level explanation for everything covered in Similar Figures. Multiplying every coordinate by k means every distance in the shape also scales by k (since distances are built from coordinate differences, and multiplying both endpoints by k also multiplies their difference by k) — which is exactly the linear scaling (k¹) from Similar Figures.
Since dilation is centered at the origin in this basic rule, the origin itself never moves (0×k = 0 always) — every OTHER point moves directly along the ray connecting it to the origin, landing at exactly k times its original distance from that center.
Using It In A Proof
Dilating From a Center Other Than the Origin
The basic (x,y)→(kx,ky) rule only works when dilating from the origin — dilating from any other center point requires an extra translation step, similar to how off-origin rotations work.
1
For origin-centered dilations, apply the rule directly
Multiply both coordinates by k.
2
For dilations from a different center (h,k_center)
Subtract the center's coordinates first (shifting the center to the origin temporarily), multiply by the scale factor, then add the center's coordinates back.
3
Full off-center formula
New point = (h + k(x−h), k_center + k(y−k_center)) — this three-step process (subtract, scale, add back) appears again in Composition of Transformations.
Full Worked Example
Dilating From a Non-Origin Center
Given: Point P(6, 4) is dilated with scale factor 2, centered at (2, 1) instead of the origin. Find: the image point.
1
Subtract the center's coordinates
(6−2, 4−1) = (4, 3) — this temporarily treats the center as the origin.
2
Multiply by the scale factor
2 × (4, 3) = (8, 6).
3
Add the center's coordinates back
(8+2, 6+1) = (10, 7).
4
Verify using distance
Distance from center (2,1) to original point (6,4): √[(6−2)²+(4−1)²] = √[16+9] = 5. Distance from center to image (10,7): √[(10−2)²+(7−1)²] = √[64+36] = 10. 10 is exactly double 5. ✓
This distance check confirms the scale factor of 2 was applied correctly relative to the actual center point, not the origin.
🎯 Quick Worked Example
A triangle with vertex (3, 5) is dilated by scale factor 1/2, centered at the origin. Find the image of this vertex.
1
Apply the rule. (x,y) → (kx, ky) with k = 1/2.
2
Calculate. (1/2 × 3, 1/2 × 5) = (1.5, 2.5).
3
Interpret. Since k = 1/2 is between 0 and 1, this is a reduction — the image point is closer to the origin than the original.
📌 Exam Application
Negative scale factors are a favorite advanced twist — a dilation with k = −2, for example, both doubles the size AND rotates the shape 180° simultaneously, since multiplying by a negative number flips both coordinates' signs (matching the 180° rotation rule) while also scaling the magnitude.
⚠️ Most Common Dilation Rules Mistakes
Trap 1 — Applying the origin-only rule to an off-center dilation: The simple (x,y)→(kx,ky) rule ONLY works when the center of dilation is the origin — using it directly for a dilation centered elsewhere gives a wrong answer; use the subtract-scale-add-back process instead.
Trap 2 — Confusing enlargement and reduction based on k: k > 1 enlarges; 0 < k < 1 reduces — it's easy to assume any k value "scales up," but a fractional k specifically shrinks the shape.
✓ Quick Self-Test
1) State the dilation rule for a scale factor k centered at the origin. 2) What range of k values produces an enlargement? What range produces a reduction? 3) Dilate the point (4, 8) by scale factor 3, centered at the origin. 4) What extra steps are needed to dilate from a center other than the origin? 5) What does a negative scale factor do to a shape, beyond just resizing it?