4.1 Dilations & Scale Factor

What a Dilation Does

A dilation is a transformation that resizes a figure without changing its shape. Every dilation is defined by two things: a fixed center of dilation (a point) and a nonzero scale factor k. The dilation stretches or shrinks the figure away from, or toward, the center by the factor k.

Because shape is preserved but size can change, a dilation is a similarity transformation, not a rigid motion. It is the one transformation on the Geometry Regents that is allowed to change size.

New York's Next Generation standard G-SRT.A.1 asks you to verify the defining properties of a dilation, and G-SRT.A.2 and G-SRT.A.3 build similarity on top of it. Mastering dilations is therefore the gateway to the entire Similarity, Right Triangles, and Trigonometry domain, the largest block on the exam at 29-37%.

The Scale Factor k

The scale factor is a ratio: k = (image length) / (preimage length). Its value tells you exactly what the dilation does.

So k = 3 triples every length, k = 1/2 halves every length, and k = -2 doubles every length while flipping the figure through the center.

Dilations on the Coordinate Plane

Centered at the origin. When the center is the origin O(0, 0), a dilation with scale factor k is simply (x, y) -> (kx, ky): multiply both coordinates by k. A scale factor of 2 maps (3, -4) to (6, -8); a scale factor of 1/2 maps (8, 6) to (4, 3).

Centered at any point. When the center C is not the origin, you cannot just multiply the coordinates. Work with the vector from the center to the point:

P' = C + k(P - C)

In words: find the horizontal and vertical distance from C to P, multiply each by k, then add the result back to C.

Worked example. Dilate P(4, 0) about center C(1, -2) with k = 3.

• Vector from C to P: (4 - 1, 0 - (-2)) = (3, 2).
• Multiply by k = 3: (9, 6).
• Add back to C: (1 + 9, -2 + 6) = (10, 4).

So P' = (10, 4). Notice that C, P, and P' are collinear: every image point lies on the ray from the center through the original point.

What a Dilation Preserves and Changes

Keep a clean mental list of what a scale factor touches and what it leaves alone.

• Angle measures: unchanged. Dilations preserve angles, which is why the image is similar to the preimage.
• Parallelism and collinearity: unchanged. A line that does not pass through the center maps to a parallel line; a line through the center maps to itself.
• Length: multiplied by |k|. Every segment, and therefore the perimeter, scales by k.
• Area: multiplied by k^2. Because area is two-dimensional, doubling lengths (k = 2) quadruples area.
• Orientation: preserved for k > 0, reversed through the center for k < 0.

A frequent Regents trap asks how the perimeter changes under a dilation. Perimeter is a length, so it scales by k, not by k^2. Only area scales by k^2. For similar solids, volume scales by k^3, so memorize the k, k^2, k^3 ladder for length, area, and volume.

Reading the Scale Factor Backward

Many questions give you the image and preimage and ask for k. Divide a pair of corresponding lengths, image over preimage. If a 6-unit segment becomes 15 units, then k = 15/6 = 5/2. If a 20-unit figure becomes 8 units, k = 8/20 = 2/5, a reduction. Always put the new figure's length on top so that k > 1 signals enlargement and k < 1 signals reduction.

Negative and Fractional Scale Factors

A fractional scale factor (0 < k < 1) produces a reduction. With k = 2/3, a triangle with sides 9, 12, 15 becomes 6, 8, 10. A negative scale factor sends each point through the center to the opposite side. Dilating A(2, 4) about the origin with k = -1 gives (-2, -4), and with k = -2 gives (-4, -8). A dilation of k = -1 about the origin has the same effect as a 180-degree rotation about the origin, a link the exam sometimes uses to blur transformation types.

The Center Is the Only Fixed Point

For any k not equal to 1, the only point that maps to itself is the center of dilation, because its distance from the center is zero and zero times k is still zero. Every other point slides along the ray from the center. This is why locating the center matters: if a figure is dilated "about vertex A," then A stays fixed and you measure all scaling from A.