L3 · Dilations on the coordinate grid (centre at the origin)

Learning goal

“I can dilate any point or polygon about the origin using the rule (x, y) -> (kx, ky), find the scale factor from a pair of corresponding vertices, and verify a dilation by the straight-line test: original vertex, image vertex and the origin lie on one straight line.”

Dilation — a transformation of size: it produces a similar figure by multiplying distances from a fixed centre by the scale factor. It is not rigid - size changes, shape does not.

Centre of dilation — the fixed point a dilation grows from or shrinks toward. In this unit the centre is ALWAYS the origin (0, 0).

Image — the figure after the transformation; its vertices are written with primes: A' (read "A prime").

Worked example

Dilate triangle A(1, 2), B(3, 2), C(1, 5) about the origin with k = 2. Step 1 (the rule): with centre (0, 0), every coordinate is multiplied by k: (x, y) -> (2x, 2y). Step 2 (vertex by vertex): A(1, 2) -> A'(2, 4). B(3, 2) -> B'(6, 4). C(1, 5) -> C'(2, 10). Step 3 (what changed, what didn't): the image is similar to the original, twice as far from the origin in every direction. The origin itself never moves: (0, 0) -> (0, 0). Say it plainly: a dilation about the origin is a MULTIPLY, applied to both coordinates. Adding k is a translation wearing a dilation's clothes.

Check yourself — explain it

Both coordinates are multiplied because...

Paired problem — example still visible

Dilate (4, -2) about the origin with k = 1/2. -> (2, -1)

Faded completion — last step is yours

"Rule (x, y) -> (3x, 3y). The vertex (-2, 1) maps to (-2 x 3, 1 x 3) = (______, ______)." -> (-6, 3)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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