Dilation Calculator
Dilate a point $(x, y)$ by scale factor $k$ from center $(c_x, c_y)$
$$ D_k(x, y) \rightarrow (x’, y’) $$
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Transformation Visualizer
Detailed Solution
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By Prof. David Anderson
Senior Math Instructor | 20+ Years Experience
“In Geometry class, transformations are a core standard, and Dilation is often the one that trips students up—especially when the center of dilation moves away from the origin or when a negative scale factor is involved. I designed this Dilation Calculator to not only show you the new coordinates but to visualize the ‘rays’ of expansion so you can intuitively understand the math behind geometric transformations.”
The Ultimate Guide to Geometric Dilations: Formulas, Scale Factors, and Coordinates
A Complete Guide for High School Geometry and College Prep
Welcome to the definitive guide on Geometric Dilation. Unlike transformations that just slide (translation), spin (rotation), or flip (reflection) a shape, a dilation changes the size of the figure while keeping its shape intact. This creates what mathematicians call similar figures. Whether you are dealing with an enlargement or a reduction, the calculation relies on two things: the Scale Factor ($k$) and the Center of Dilation.
⚠️ Disambiguation: Medical vs. Math
Note: Are you looking for pupil dilation or cervical dilation? This page features a Geometry Calculator for the coordinate plane.
• Dilation (Geometry): A transformation that changes the size of a figure.
• Dilation (Medical): The widening of an opening (like the eye).
If you are a student solving for scale factor geometry problems, you are in the right place.
1. The Dilation Formulas: Origin vs. Non-Origin
The most common mistake students make is applying the “Origin Formula” when the center of dilation is not $(0,0)$. We must distinguish between the two scenarios to correctly use the Dilation Calculator.
Scenario A: Center is Origin (0,0)
$$ P'(kx, ky) $$
Simply multiply the coordinates by the scale factor $k$.
Scenario B: Center is $(a, b)$
$$ x’ = a + k(x – a) $$
$$ y’ = b + k(y – b) $$
Subtract center $\to$ Scale $\to$ Add center back.
Prof. Anderson’s Strategy
Think of Scenario B as a “Shift-Scale-Shift” maneuver.
1. Shift the world so the center $(a,b)$ becomes the origin.
2. Scale the point using the easy dilation formula.
3. Shift the world back to original position.
2. Deep Dive: Understanding Scale Factor ($k$)
The Scale Factor ($k$) determines the fate of your shape. It tells us how much the pre-image (original) will grow or shrink to become the image (result).
| Value of $k$ | Type of Dilation | Visual Effect |
|---|---|---|
| $k > 1$ | Enlargement | Shape gets larger and moves away from center. |
| $k = 1$ | Identity | No change. The shape stays exactly the same. |
| $0 < k < 1$ | Reduction | Shape gets smaller and moves closer to center. |
| $k < 0$ (Negative) | Rotation + Scale | Shape flips 180° around the center AND scales by $|k|$. |
3. Step-by-Step: How to Dilate a Triangle
Often, homework asks to “Dilate Triangle ABC.” To do this, simply use our Dilation Calculator on each vertex ($A, B, C$) individually. This works for any polygon on the coordinate plane.
Step 1
List Coordinates
Write down the $(x, y)$ for all three points of the triangle.
Example: $A(2,4), B(4,4), C(3,6)$.
Example: $A(2,4), B(4,4), C(3,6)$.
Step 2
Apply Formula
Multiply each coordinate pair by the scale factor $k$ (if center is origin).
If $k=2$: $A'(4,8), B'(8,8), C'(6,12)$.
If $k=2$: $A'(4,8), B'(8,8), C'(6,12)$.
Step 3
Verify Distances
The side lengths of the new triangle should be $k$ times the original.
Length $A’B’ = k \cdot \text{Length } AB$
4. Advanced: Negative Scale Factors
What happens if $k = -2$? This is a favorite trick question on geometry exams. A negative scale factor dilation performs two actions simultaneously.
First, it creates a point reflection (rotation of 180°). The image appears on the opposite side of the center of dilation. Second, the distance is determined by the absolute value $|k|$.
Example: Point $P(2, 3)$, Center $(0,0)$, $k = -1$.
The new point is $P'(-2, -3)$. This is effectively a rotation of 180 degrees around the origin without changing size.
5. Real-World Applications
- 📸 Photography & Graphic Design: Resizing an image on a screen is a geometric dilation. The “Center of Dilation” is usually the anchor point (e.g., top-left corner or center) you drag from.
- 🔭 Ophthalmology: While this tool is for geometry, the math of pupil dilation follows similar principles of increasing aperture area to allow more light (scaling the radius).
- 🗺️ Cartography (Maps): Creating a “Zoomed In” map inset is a dilation. The scale on a map (e.g., 1:1000) is literally the inverse of the scale factor $k$.
6. Practice Corner: Test Your Knowledge
📝 Practice Problem
Problem: Dilate Point $M(5, -2)$ by a scale factor of $k = 3$ with the center of dilation at $C(1, 2)$. Find $M’$.
Solution:
1. Calculate horizontal distance: $5 – 1 = 4$. Scale it: $4 \times 3 = 12$. Add to center: $1 + 12 = 13$. ($x’ = 13$)
2. Calculate vertical distance: $-2 – 2 = -4$. Scale it: $-4 \times 3 = -12$. Add to center: $2 + (-12) = -10$. ($y’ = -10$)
Answer: $M'(13, -10)$.
7. Professor’s FAQ Corner
Can I dilate a line segment?
Yes. To dilate a line segment, simply dilate its two endpoints using our Dilation Calculator. The new line segment will be parallel to the original (unless the center of dilation lies on the line itself) and its length will be multiplied by $k$.
What if the center of dilation is ON the shape?
This is fine! The points that are exactly on the center of dilation do not move (they are “invariant points”). The rest of the shape expands or shrinks around that fixed point.
Is a Dilation a “Rigid Transformation”?
No. Rigid transformations (isometries) like translation, rotation, and reflection preserve size. Dilation creates Similar Figures (same angles, proportional sides) but not Congruent Figures (unless $k=1$).
References & Further Reading
- Jurgensen, R., & Brown, R. (2000). Geometry. McDougal Littell. (Chapter 9: Transformations).
- Khan Academy. “Dilations: Introduction & center of dilation.” Comprehensive video modules.
- Math Open Reference. “Dilation of a polygon.” Interactive geometry tools.
- Wolfram MathWorld. “Homothety.” (The formal mathematical term for dilation).
Solve Your Coordinate Problem
Stop struggling with the $x’ = a + k(x-a)$ formula. Enter your points above and see the geometric transformation instantly.
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