Dilation
In short: Dilation resizes a figure about a fixed center using a scale factor: angles and overall shape stay, while every length is multiplied by that same factor.
8th GradeGeometry8.G.A.3
What is Dilation?
Think of a projector: a dilation pushes a figure outward or pulls it inward along rays from a center point. With scale factor \(k\), each image point lies on the ray through the original point, at \(k\) times the original distance from the center.
Around the origin, \((x, y)\) turns into \((kx, ky)\). Values over 1 enlarge, fractions between 0 and 1 shrink, and exactly 1 leaves things alone. Angles remain equal and each side stays parallel to its image, so the result is similar to the figure you started with, though usually not congruent.
How to spot it
- The wording says enlarge, reduce, stretch or scale a figure by a factor from a given center.
- Coordinates come with a rule like (x, y) → (2x, 2y).
- A drawing shows two copies of one shape, one larger, with lines from a common point through matching corners.
- A photo, map or model is resized while its proportions are protected.
The method
- Note the center of the dilation and the factor \(k\).
- With the center at the origin, multiply both coordinates of each corner by \(k\).
- Mark the new corners and join them in order; matching sides must come out parallel.
- Check lengths: every image side should be \(k\) times its original partner.
Dilation: solved example
Problem
A small triangle ABC has corners \(A(1, 1)\), \(B(1, 3)\) and \(C(4, 1)\). Blow it up from the origin by a factor of 2, then name the new corners and compare the areas.
Solution
Double each coordinate: \(A^{\prime} = (2, 2)\), \(B^{\prime} = (2, 6)\), \(C^{\prime} = (8, 2)\).
The original triangle has legs 2 and 3, so its area is \(\frac{1}{2}(2)(3) = 3\). The image has legs 4 and 6 and area \(\frac{1}{2}(4)(6) = 12\), exactly \(2^2 = 4\) times more. The new corners are \((2, 2)\), \((2, 6)\) and \((8, 2)\).
Try it
Question 1
Dilate the point \((6, 8)\) from the origin with scale factor \(\frac{1}{2}\).
Show the answer
\(\left(\frac{1}{2} \cdot 6, \frac{1}{2} \cdot 8\right) = (3, 4)\), a reduction.
Question 2
An edge of 9 cm grows to 15 cm under this kind of resizing. Which multiplier did it use?
Show the answer
\(\frac{15}{9} = \frac{5}{3}\), an enlargement.
Dilation vs. Congruent Figures
This transformation changes size, so the image is similar to the original but usually not congruent. Slides, flips and turns keep size and shape, so their images are congruent. Only a factor of 1 or negative 1 gives a congruent image.
Keep going
Lessons
Worksheets and quizzes
- Dilations and Similar Figures – Worksheet
- Dilations and Similar Figures – Quiz
- Dilations and Similarity – Worksheet
- Dilations and Similarity – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry