Dilation

Dilation is a transformation that changes the size of a figure but not its shape. When a figure is dilated, all of its points move away from or toward a fixed point called the center of dilation. The distance from each point to the center of dilation is multiplied by the same scale factor in a dilation.

Key Concepts:

1. Center of Dilation: The fixed point about which the figure is dilated.
2. Scale Factor: The constant factor by which all distances from the center of dilation are multiplied. A scale factor greater than 1 results in an enlargement, while a scale factor between 0 and 1 results in a reduction.
3. Dilated Image: The resulting figure after dilation.

Properties of Dilations:

1. Corresponding angles are congruent.
2. Corresponding side lengths are in proportion to the scale factor.
3. The image and pre-image are similar figures.

Example:

Let's say we have a triangle ABC with a center of dilation at point O and a scale factor of 2. The coordinates of the vertices are A(1, 2), B(3, 4), and C(5, 6). To dilate the triangle with respect to point O and a scale factor of 2, we would multiply the distance from each vertex to point O by 2 to get the new coordinates of the dilated triangle.

Study Guide:

1. Understand the concept of dilation and how it differs from other transformations like translation, rotation, and reflection.
2. Learn to identify the center of dilation and the scale factor when given a dilation problem.
3. Practice calculating the new coordinates of dilated figures using the center of dilation and scale factor.
4. Understand the properties of dilations and how they affect angles, side lengths, and similarity of figures.
5. Work on various problems involving dilations to strengthen your understanding of the concept.