Track a dilation beyond the origin

Try

Put the center of enlargement at O=(1,−1), not at the coordinate origin. A dilation of scale 2 sends P=(2,0) to P′=(3,1): travel from O to P, then continue twice that displacement from O. It does not send P to (4,0), because multiplying coordinates directly assumes a center at (0,0).

A positive dilation of scale k sends any point X to O+k(X−O). The vector X−O means the horizontal and vertical displacement from the center. In coordinates, x′=oₓ+k(x−oₓ) and y′=oᵧ+k(y−oᵧ). The symbols oₓ and oᵧ are the center’s coordinates, not extra adjustable scale factors.

Predict

Predict the image of Q=(4,1) under the same dilation. The displacement from O is (3,2), which doubles to (6,4); adding O gives Q′=(7,3). The original segment displacement Q−P=(2,1) becomes Q′−P′=(4,2). Both components double, so the segment stays parallel to its image and its length doubles.

Will the line y=x+1 map to itself? It does not pass through O: at x=1 that line has y=2, not −1. Expect a distinct parallel line. A line through the center behaves differently because extending each center-to-point ray keeps every image point on that same line.

Experiment

Use the coordinate table to construct two original points and their images. Draw the original line through U and V, then draw the image line through U′ and V′. Keep the coordinate scale the same in both directions.

Point	Original coordinates	Displacement from O	Image under scale 2
U	(0,1)	(−1,2)	(−1,3)
V	(1,2)	(0,3)	(1,5)
P	(2,0)	(1,1)	(3,1)
Q	(4,1)	(3,2)	(7,3)

The first two rows concern y=x+1. The last two concern a different segment; do not place all four original points on the same line. Compute both segment lengths with the distance formula and compare image/original ratios. Your construction should show the center-to-point alignment as well as the final image.

The live explorer below uses a different fixed segment, P=(−2,−1) and Q=(2,1), on the line x−2y=0. Its center and scale are editable. Move the center onto that line and then away; compare scale 1 with another positive scale. Predict whether the full image line is identical or distinct and parallel, then inspect the line equation and segment-length ratio. Coordinates outside the fixed viewing window still belong to the mathematical construction.

Dilate a whole line from a chosen center

The source segment joins P=(−2,−1) to Q=(2,1) on the line x−2y=0. Choose a center C and positive scale k. Predict the image endpoints and whether the full image line is the same line or a distinct parallel line. The third line is the parallel line through C.

Center x-2-1012Center y-2-1012Positive scale k1/213/223Reset dilationReveal dilation results

C=(1,-1); k=2. The source line is dashed; its marked segment is thick. The line through C is dotted. After revealing, the image line is solid red and its image segment is thicker. Coordinates outside the graph remain part of the construction.

Endpoint calculation X′=C+k(X−C)

Point	Subtract center	Scale and add center
P	(-3,0)	Predict
Q	(1,2)	Predict

Why this describes every point on the line

For any source point X=(x,y), write X′=C+k(X−C). Then x′−2y′=k(x−2y)+(1−k)(Cₓ−2Cᵧ). On the source line the first term is zero, giving the displayed image equation. Conversely, any point on that image line comes from X=C+(X′−C)/k. Since k is nonzero, this inverse exists. The image is the entire line, not just the two displayed endpoints.

On the line through C, x−2y=Cₓ−2Cᵧ. Substitution gives the same constant after dilation. Positive k places each point on the same ray from C; k=0 would collapse everything to C and is excluded.

Try a center on the source line and a center away from it, then compare k=1/2, 1 and 2. Record the construction and the general reason in practice. The fixed axes never rescale, and temporary controls do not count as mastery.

Understand

For any two points X and Y, subtract their image formulas: Y′−X′=k(Y−X). The center terms cancel. When k>0, the new displacement has the same direction and length multiplied by k. The distance statement follows by squaring its coordinate components: k² times the old squared distance gives k times the old nonnegative distance.

A whole line can be written X=P+t v, where v is a fixed nonzero direction and t runs through all real numbers. Its image is O+k(P−O)+kt v. Because kt also runs through every real number, the image is an entire parallel line, not a finite segment. If O lies on the original line, the image is that same line. If O does not lie on it and k≠1, the lines are distinct. For k=1 every point stays fixed, including off-center lines; that exception matters.

For a nonvertical line y=mx+b, substitute the coordinate formulas to obtain

y′=mx′+kb+(1−k)(oᵧ−m oₓ).

With m=1, b=1, O=(1,−1) and k=2, the new intercept is 2+(−1)(−2)=4. Thus the image is y′=x′+4. Both U′ and V′ satisfy it. A vertical line x=c instead maps to x′=oₓ+k(c−oₓ); it should not be forced into a finite-slope formula.

Positive dilations preserve angles because every ray direction is preserved after moving its vertex to the image position. Their side lengths share one common scale factor. This uses Euclidean angle measurement and the parallel-angle results, not the AA similarity theorem that we will derive next.

Apply

Construct an image under a fractional scale between zero and one, then verify a line and a segment. Fractional positive scales shrink distances toward the center but do not reverse the center-to-point direction. The center itself remains fixed for every scale.

A false argument says that a dilation adds a constant to every length. Our segment displacement proof shows multiplication instead: a segment of length 2 and one of length 5 become 2k and 5k. Their increases need not be equal. Another false argument multiplies coordinates while ignoring an off-origin center; repair it with subtract, scale, add.

The written construction should name O, k and each correspondence, not just draw a larger-looking shape.

Continue

A similarity witness can combine a dilation with translations, rotations and reflections. Positive dilation changes size uniformly; the rigid motions change placement or orientation without changing size. The next lesson builds and verifies such a witness.

Optional depth: k=0 collapses the plane to O and is not an invertible similarity. Negative scales combine a positive enlargement by ∣k∣ with a half-turn about O. The school constructions here use k>0 so that corresponding center rays are unambiguous.