The student will:

• identify the properties of dilatations.

source:  transformation

• The object and image are similar (~) when the dilatation factor  ¹ 1. (corresponding angles are congruent).
• The object and image are congruent when the dilatation factor = 1. (corresponding angles and sides are congruent).

mapping: (x, y) --> (3x, 3y)

dilatation centre = origin

(x, y) --> (3x, 3y)

DABC ~ DA'B'C'

DABC is being enlarged by a factor of 3 relative to the origin.

A(2,3), B(2,1), C(1,1)

Image DA'B'C' is

A'(6, 9), B'(6, 3), C'(3, 3)

Vocabulary

dilatation - a transformation that changes the size of an object

enlargement - a dilatation where the image is larger than the original object

reduction- a dilation where the image is smaller than the original object

scale factor can be calculated from the ratio:

length from origin to image

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length from origin to original

dilatation centre - lines drawn through corresponding image vertices will meet at the dilatation centre

coordinates - an ordered pair of the form (x, y) that locates a point on a coordinate plane

corresponding sides - sides that have the same relative positions in geometric figures

proportional - two objects are proportional if the all the ratios of the corresponding sides are the same

perspective - the different views of an object - top, bottom, side, front

vanishing point - is the point at which, if two parallel lines, or walls were extended into the distance as far as you could see, would look like they meet or vanish into the distance. A great example of this is railway tracks. When you look down the tracks, it looks like the tracks eventually come together.

JavaSketchpad: Playing with Perspective - Sanders

Playing Around with Size and Distance

Exploring Linear Perspective - includes vanishing point

The Tutorial leads you through a series of dilations involving the same original triangle. You will discover that dilations can be mapped in the same way as congruence transformations, for example, (x,y) --> (2x, 2y). You will learn that:

- to find the coordinates of the image vertices, multiply the coordinates of the object vertices by the scale factor;

- to find the length of each side on the image, multiply the corresponding side of the object by the scale factor;

- lines drawn through corresponding image vertices will meet at the dilation centre;

- the distance from the dilation centre to the image can be determined by:

distance of object from dilation centre X scale factor;

- a scale factor greater than 1 produces an enlargement

- a scale factor between 0 and 1 produces a reduction

The Learning Equation 9 Teacher's Manual, page 192.