For an intuitive review of dilations, see the Refresher section Transformations: Dilations.
Δ
What happens when scale factor
Let's see how a negative dilation affects a triangle:
Did you notice the connection between negative dilations and rotations? When k = -1, the triangles are congruent and the negative one scale factor produces the exact same result as a rotation of 180º centered at O. We can write: .This direct connection between dilations and rotations does NOT extend to dilations whose scale factors do NOT equal -1, such as the dilations ofk = -½ and k = -2. These dilations do not preserve length, and rotations must preserve length. We can, however, express these dilations as a composition of transformations involving a rotation. For example, when k = -2, we can also describe that transformation as a dilation of scale factor +2 combined with a rotation of 180º (both centered at O). We can write:
Dilations on Coordinate Axis: Dilation with Center at Origin, (0,0): Formula: Center at Origin:
Dilation with Center NOT at Origin:
The counting of vertical and horizontal distances shown above is a simple and easy way to find the coordinates for a dilation not centered at the origin.
• First translate the center of the dilation so the origin becomes the center. • Then apply the dilation factor, • And translate back: ( Formula: Center Not at Origin:
Let ( Rule: (x,y) → (3x - 8, 3y + 4) For example, under this dilation, the point (5,6) becomes (3(5)-8, 3(6)+4) which is (7,22).
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# Dilations - MathBitsNotebook(Geo - CCSS Math) (2023)

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