Dilations: Step-by-Step Guide
A dilation makes a shape bigger or smaller by multiplying all distances from a center point by a scale factor k. When the center is the origin, a point (x, y) becomes (kx, ky). If k is bigger than 1, the shape gets bigger. If k is between 0 and 1, it gets smaller. The new shape has the same angles, so it is similar to the original.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Find the center and the scale factor. Find the center of the dilation (usually the origin) and the scale factor k.\[\text{center} = (h, j),\; \text{scale factor} = k\]
- Use the rule (center at the origin). Multiply both coordinates of every point by k.\[(x, y) \mapsto (kx,\; ky)\]
- If the center is not the origin. Move everything so the center is at the origin, dilate, and move it back. The result is (x, y) becomes (h + k(x - h), j + k(y - j)), where (h, j) is the center.\[(x, y) \mapsto (h + k(x - h),\; j + k(y - j))\]
- Find the new points. Work out each new point and label it with a prime, like A'.\[A(x_A, y_A) \mapsto A'(kx_A, ky_A)\]
- Check the side lengths. Each side of the new shape should be |k| times as long as the matching side of the original.\[\frac{|A'B'|}{|AB|} = |k|\]
Worked example
Triangle ABC has vertices A(2, 3), B(4, 1), C(6, 5). Apply a dilation centered at the origin with scale factor k = 2 and find the image vertices.
Step 1: The center is the origin (0, 0) and the scale factor is k = 2.
Step 2: The rule is (x, y) becomes (2x, 2y).
Step 3: A(2, 3) becomes A'(2 x 2, 2 x 3) = A'(4, 6).
Step 4: B(4, 1) becomes B'(2 x 4, 2 x 1) = B'(8, 2).
Step 5: C(6, 5) becomes C'(2 x 6, 2 x 5) = C'(12, 10).
Check: Side AB has length the square root of 8. Side A'B' has length the square root of 32, which is 2 times the square root of 8. The ratio is 2, which matches k.
The new vertices are A'(4, 6), B'(8, 2), and C'(12, 10).
Tips and common mistakes
- When the center is the origin, just multiply each coordinate by k.
- If k is negative, the shape is also flipped to the other side of the center.
- If k is bigger than 1, the shape gets bigger. If k is between 0 and 1, it shrinks. If k = 1, nothing changes.
- A dilation keeps the angles and keeps parallel lines parallel. The new shape is similar to the original, but not congruent (unless k = 1 or -1).
- Side lengths are multiplied by |k|. Areas are multiplied by k^2.
- If the center is not the origin, move the center to the origin first, dilate, and then move back.
- Each new point lies on the line from the center through the original point.
- Label new points with primes (A') and double-check your multiplication.