What Is a Dilation in Geometry?
A dilation in geometry is a transformation that changes the size of a figure without changing its shape. It is one of the four basic transformations, alongside translation (slide), reflection (flip), and rotation (turn), but it is the only one of the four that changes size. Every dilation is fixed by two things:
The center of dilation, the single point that stays put while everything else moves toward or away from it.
The scale factor (k), the number that says how many times bigger or smaller the image becomes.
Because every point moves along a straight line from the center, and every distance is multiplied by the same k, the image is similar to the original: same angles, same proportions, different size. A dilation with $k > 1$ is an enlargement (the figure grows); a dilation with $0 < k < 1$ is a reduction (the figure shrinks); and $k = 1$ leaves the figure unchanged. The scale factor cannot be zero, that would collapse the whole figure to the single center point.
How Do You Perform a Dilation on the Coordinate Plane?
When the center of dilation is the origin, the rule is as clean as it gets. Multiply every coordinate by the scale factor:
$$(x, y) ;\longrightarrow; (kx, ky).$$
So under a scale factor of 3 about the origin, the point $(2, 5)$ becomes $(6, 15)$. That is the whole procedure for an origin-centred dilation, scale each coordinate, every time.
Why multiplying the coordinates works
This is worth seeing once. A point $(x, y)$ sits at a horizontal distance x and a vertical distance y from the origin. A dilation about the origin multiplies the distance from the center by k, and since the center is the origin, the horizontal distance becomes $kx$ and the vertical distance becomes $ky$. The image point is therefore $(kx, ky)$, the coordinate rule is not a separate fact, it is what "multiply the distance from the center by k" means when the center is $(0, 0)$.
When the center is not the origin
If the center is some other point $(a, b)$, you cannot just multiply the raw coordinates, you have to measure distance from the center first. The rule becomes:
$$(x, y) ;\longrightarrow; \big(a + k(x - a),; b + k(y - b)\big).$$
In words: subtract the center to find how far the point sits from it, scale that gap by k, then add the center back. (The origin version is just this rule with $a = b = 0$.)
How Do You Find the Scale Factor of a Dilation?
The scale factor is a ratio: the size of the image divided by the size of the original.
$$k = \frac{\text{image length}}{\text{original length}}.$$
Pick any length on the original figure and the matching length on the image, a side, a radius, the distance from the center to a vertex, and divide. If a square's side grows from 5 to 20, then $k = 20 / 5 = 4$. If a circle's radius shrinks from 8 to 2, then $k = 2 / 8 = \tfrac{1}{4}$, a reduction. The same ratio holds for every corresponding pair of lengths, which is exactly what makes the figures similar.
To find the center of a dilation when it is not given, draw a line through each original vertex and its image; all those lines meet at one point, and that point is the center.
Examples of Dilation in Geometry
With the coordinate rule and the scale factor in hand, here are dilations being carried out. The problems move from a single point about the origin up to a non-origin center and a negative scale factor.
Example 1 - Dilate the point $A(3, 4)$ about the origin with scale factor $k = 2$
Multiply both coordinates by 2:
$$A(3, 4) ;\longrightarrow; A'(2 \cdot 3,; 2 \cdot 4) = A'(6, 8).$$
Final answer: $A'(6, 8)$. The image sits twice as far from the origin as A, along the same ray.
Example 2 - A triangle has vertices $P(2, 2)$, $Q(4, 2)$, $R(2, 6)$. Dilate it about the origin with scale factor $k = \tfrac{1}{2}$
Wrong attempt. A student sees "scale factor one-half" and divides each coordinate by $\tfrac{1}{2}$, getting $P'(4, 4)$, $Q'(8, 4)$, $R'(4, 12)$, a bigger triangle. But $k = \tfrac{1}{2}$ is between 0 and 1, so this should be a reduction: the image must be smaller, not larger. Dividing by a fraction multiplies, which is the opposite of what a reduction does.
Correct. The rule is always multiply by k, never divide:
$$P(2,2) \to P'(1, 1), \quad Q(4, 2) \to Q'(2, 1), \quad R(2, 6) \to R'(1, 3).$$
Final answer: $P'(1, 1)$, $Q'(2, 1)$, $R'(1, 3)$, a triangle half the size, same shape.
Example 3 - A square has side length 6 cm. After a dilation its side is 15 cm. Find the scale factor
Scale factor is image over original:
$$k = \frac{15}{6} = 2.5.$$
Final answer: $k = 2.5$. Since $k > 1$, this is an enlargement.
Example 4 - Dilate the point $B(5, 3)$ with center $C(1, 1)$ and scale factor $k = 3$
Use the non-origin rule. First find the gap from the center: $(5 - 1,; 3 - 1) = (4, 2)$. Scale it by 3: $(12, 6)$. Add the center back:
$$B'(1 + 12,; 1 + 6) = B'(13, 7).$$
Final answer: $B'(13, 7)$.
Example 5 - A photo 4 inches wide is enlarged so that its width becomes 10 inches. By what scale factor, and what is the new height if the original was 6 inches tall?
The scale factor comes from the widths:
$$k = \frac{10}{4} = 2.5.$$
Height scales by the same factor: $6 \times 2.5 = 15$ inches.
Final answer: $k = 2.5$, new height 15 inches. The proportions stay fixed, which is exactly why a dilated photo never looks stretched.
Example 6 - Dilate the point $D(4, 2)$ about the origin with scale factor $k = -2$
A negative scale factor does two things at once: it scales by $|k| = 2$ and sends the image to the opposite side of the center (a 180° point reflection). Multiply each coordinate by $-2$:
$$D(4, 2) ;\longrightarrow; D'(-8, -4).$$
Final answer: $D'(-8, -4)$. The image is twice as far from the origin as D, but in the opposite direction.
Why Dilation Matters
A dilation is not just a worksheet move, it is the formal name for "resize without distorting," and that operation runs through a surprising amount of the world.
Maps and scale models. A map is a dilation of the real world with a scale factor like $\tfrac{1}{50000}$. Every distance shrinks by the same ratio, which is why a straight road on the ground stays straight on the map and the angles between streets are preserved.
Image scaling on screens. When you pinch-zoom a photo or a webpage scales for a larger monitor, the device performs a dilation about the point under your fingers. The shape stays true because every pixel's distance from the center is multiplied by the same factor.
Similarity and the foundation of trigonometry. Two figures are similar exactly when one is a dilation (possibly with a rotation or reflection) of the other. This is the bedrock under similar triangles, and similar triangles are why the trig ratios work the same for a 30° angle whether the triangle is tiny or enormous.
Blueprints and manufacturing. An architect's drawing is a reduced dilation of a building; a microchip mask is an enlarged dilation of features later shrunk in fabrication.
For a Grade 9 student, dilation is where "similar shapes" stops being a vocabulary word and becomes a precise instruction: a figure is similar to another when a single center and a single scale factor turn one into the other.
Where Things Go Sideways With Dilation
Mistake 1: Dividing by a fractional scale factor instead of multiplying
Where it slips in: The scale factor is between 0 and 1 and the student reads "shrink" as "divide."
Don't do this: Divide each coordinate by $\tfrac{1}{2}$ for a reduction.
The correct way: Always multiply by k. A scale factor of $\tfrac{1}{2}$ multiplies each coordinate by $\tfrac{1}{2}$, which halves it. Check the result: a reduction must come out smaller than the original.
Mistake 2: Ignoring the center when it is not the origin
Where it slips in: The center is $(a, b) \neq (0, 0)$ and the student multiplies the raw coordinates anyway.
Don't do this: Apply $(kx, ky)$ when the center is $(1, 1)$.
The correct way: Measure from the center first. Subtract the center, scale the gap by k, then add the center back: $\big(a + k(x - a),; b + k(y - b)\big)$.
Mistake 3: Assuming a dilation changes the angles
Where it slips in: A student thinks "bigger figure" means "different shape."
Don't do this: Recompute or change the angles after enlarging.
The correct way: A dilation preserves every angle and every proportion. Only lengths change, and they all change by the same factor. The image is always similar to the original.
Key Takeaways
A dilation in geometry resizes a figure about a center by a scale factor, changing size but keeping shape, so the image is always similar to the original.
About the origin the rule is $(x, y) \to (kx, ky)$; about a center $(a, b)$, measure from the center first.
The scale factor is image length over original length: $k > 1$ enlarges, $0 < k < 1$ reduces, and $k$ cannot be 0.
A negative scale factor resizes and flips the figure to the opposite side of the center.
The most common mistake is dividing by a fractional scale factor instead of multiplying, always multiply, then check the image came out the expected size.
Practice These Problems to Solidify Your Understanding
Dilate the point $A(6, 9)$ about the origin with scale factor $k = \tfrac{1}{3}$.
A triangle's side grows from 4 cm to 18 cm after a dilation. Find the scale factor.
Dilate the point $B(7, 4)$ with center $C(3, 2)$ and scale factor $k = 2$.
Answer to Question 1: $A'(2, 3)$, since $\tfrac{1}{3} \times 6 = 2$ and $\tfrac{1}{3} \times 9 = 3$. Answer to Question 2: $k = 18 / 4 = 4.5$, an enlargement. Answer to Question 3: $B'(11, 6)$; the gap from the center is $(4, 2)$, scaled by 2 is $(8, 4)$, plus the center gives $(11, 6)$. If Question 1 gave you a bigger point, recheck whether you multiplied or divided (see Mistake 1).
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