G.SRT.A.1STUDENT NOTES & PRACTICE WS #5–geometrycommoncore.com1

There are different ways to alter the shape using scalar multiplication. If we scale all dimensions by the same value we get a proportional shape based off of dilation. If you scale the dimensions of the shape differently we get what is commonly called a stretch. It gains that name because the image is now distorted in some way because not all things have been enlarged or reduced at the same value. Here are some examples:

Original / Dilated (Proportional) / Dilated (Proportional)

Dilations have coordinate rules that look like . Notice that each dimension of the 2-D shape is being scaled with the exact same value, 3. This is dilation.

Original / Stretched (Distorted) / Stretched (Distorted)

Stretches have coordinate rules that look like . Notice that the dimensions of the shape are being altered by different scale values – this distorts or stretches the shape. This is a stretch.

Determine whether the following are stretch or dilation transformations.
Stretch
(Scalar Values Different) / Dilation
(Shape is Proportional) / Stretch
(Shape is Distorted)
NYTS (Now You Try Some)
1. Determine whether the following are stretch or dilation transformations.
a) / b) / c)
Stretch or Dilation / Stretch or Dilation / Stretch or Dilation

The coordinate rule for dilation

There is not much to discover here because the coordinate rule is provided in the definition,

/ When O is the origin

We had already discovered this intuitively in G.CO.2 when we were investigating isometric and non-isometric transformations. We found out that if you multiply one or two variables by a value other than 1 or -1 you are no longer isometric. We also established a difference between the dilation and the stretch. A dilation must have the same value multiplied to both variables, whereas the stretch has different values.

DILATION / STRETCH

For a dilation to maintain its proportionality of sides, the two variables
must be multiplied by a constant value, k, known as the scale factor.

COORDINATE RULE OF DILATION WHEN THE CENTER IS THE ORIGIN (0,0)

A dilation of 2 with center of dilation O, the origin.

A key to understanding dilations from the origin is that the coordinates
of the point being dilated represent the slope values of (run,rise).

When we dilate the point C(1,3) by a scale factor it has a rise of 3 and a run of 1 from the origin and that when multiple it by the scale factor of 2, we are actually doubling that slope to a rise of 6 and a run of 2. Slope is a major understanding to dilation.

A dilation of ½ with center of dilation O, the origin.
Notice again how when dilating from the origin the coordinates of the points represent a slope. So in the case of A(6,-4) is a rise of -4 and a run of 6 and so when we dilate by we want half of that slope, A’(3,-2). The same occurred for the point B (2,-8), when we dilate by we get half of the slope, B’(1,-4).
A dilation of -1/3 with center of dilation O, the origin.
Slope again is found here in this dilation even though it is a negative scale factor. The negative make the slope the exact opposite. So as you can see point D (3,9) with a rise of 9 and a run of 3 maps to D’(-1,-3) because the negative reverse the slope to a rise of -9 and a run of -3 and then the 1/3 reduces it down to a rise of -3 and a run of -1.
NYTS (Now You Try Some)
2. Dilate the following. (O is the origin).

a) b) c)d) e) f)

COORDINATE RULE OF DILATION WHEN THE CENTER IS NOT AT THE ORIGN (0,0)

A dilation of 2 with the center of dilation at T (-3,4).

/ Rise and Run Technique
To dilate by a scale factor of 2 we double the distance,
To do double the distance we will double
the rise and run from the center of dilation.

So to double everything we would go
up 4 (scale factor times 2) and
right 10 (scale factor times 5).
from the center of dilation T(-3,4).


We don’t simplify this slope in anyway because
it is actually describing the VECTOR <5,2>.
A dilation of ¼ with the center of dilation at T (8,1).

/ Rise and Run Technique
To dilate ¼ the distance, we do ¼ the rise and run
from the center of dilation.

So to ¼ of everything we would go
up 2 (scale factor times 8) and
left 3 (scale factor times -12).
from the center of dilation T(8,1).


We don’t simplify this slope in anyway because
it is actually describing the VECTOR <-12,8>.

NYTS (Now You Try Some)

3. Complete the following. (When calculating the slope do not simplify it in any way!! The slope is actually a vector.)
a) Center of dilation is G. G (1,3) A (6,5)
Scale Factor 2 / b) Center of dilation is G. G (2,5) A (0,1)
Scale Factor 3
Determine the slope of from to
/ Determine the slope of from to

Determine A’.
(____ + (2)(____) , ____ + (2)(____)) = A’ (____ ,____) / Determine A’.
(____ + (3)(____) , ____ + (3)(____)) = A’ (____ ,____)

Here is the general relationship for all dilations centered at (a,b) with a scale factor of k.