G.SRT.1 NOTES – PATTERSON 1

CONCEPT 1 – The properties and characteristics of dilations.

Dilation is the term that we use quite regularly in the English language to describe the enlarging or shrinking of our pupils. Pupils dilate either larger or smaller depending on the amount of light that enters the eye. This real world example helps us to understand the use of dilation in geometry as well – dilation is a transformation that produces an image that is the same shape as the pre-image but is a different size, either larger or smaller.

When we make things bigger using dilation we refer to that as an expansion or enlargement whereas if we use a dilation to make something smaller we use the term contraction or reduction.

DILATION – CONTRACTION (REDUCTION) / DILATION – EXPANSION (ENLARGEMENT)
PRE-IMAGE / IMAGE / PRE-IMAGE / IMAGE

When the dilation is an enlargement, the scale factor is greater than 1. What this means is that if you have a scale factor of 3, or sometimes written as 1:3, the image is three times bigger proportionally to the pre-image.

When the dilation is a reduction, the scale factor is between 0 and 1. What this means is that if you have a scale factor of ½, or sometimes written as 2:1, that the image is half the size proportionally to the pre-image.

So what happens if the scale factor is 1? Nothing!! To multiply anything by 1 maintains the pre-image. It is like rotating a shape 360° - the shape is not altered in any way – it is an identity transformation.

So what actually happens when a shape is dilated?
The length of each side of the image is equal to the length of the corresponding side of the pre-image multiplied by the scale factor, A’B’ = k · AB, B’C’ = k · BC and A’C’ = k · AC.
This dilation has a scale factor of 1:3.
AB: A’B’ 4: 12 1:3
BC: B’C’ 5.5: 16.5 1:3
AC: A’C’ 5:15 1:3 /
The distance from the center of the dilation to each point of the image is equal to the distance from the center of the dilation to each corresponding point of the pre-image figure times the scale factor,
OB’ = k · OB and OC’ = k · OC.
Scale Factor of 1:k
OB = y OB’ = ky y:ky 1:k
OC = x OC’ = kx x:kx 1:k /

DEFINITION

A dilation with center O and a scale factor of k is a transformation that maps every point P in the plane to point P’ so that the following properties are true.
1. If P is NOT the center O, then the P’ lies on .
The scale factor k is a positive number such that k = and k ¹ 1. / 2. If P is the center point O, then P = P’. The center of dilation is the only point in the plane that does not move.
/ > 1
/ 0 < < 1
/
NOTATION
/ O is the center of dilation.
k is the value of the scale factor.

PROPERTIES

DILATION PROPERTIES - A dilation is NOT an isometric transformation so its properties differ from the ones we saw with reflection, rotation and translation. The following properties are preserved between the pre-image and its image when dilating:

·  Angle measure (angles stay the same)

·  Parallelism (things that were parallel are still parallel)

·  Collinearity (points on a line, remain on the line)

·  Distance IS NOT preserved!!!

After a dilation, the pre-image and image have the same shape but not the same size.

TRANSFORMATION PROPERTIES – The following properties are present in dilation:

·  DISTANCES ARE DIFFERENT (PROPORTIONAL) – The distance points move during dilation depend on their distance from the center of dilation - points closer to the center of dilation will move a shorter distance than those farther away. In our example and point B is farther away from the center of dilation O than point P, thus .

·  ORIENTATION IS THE SAME – The orientation of the shape is maintained.

·  SPECIAL POINTS – The center of dilation is an invariant point and does not move in a dilation. If the pre-image (P) = image (P’) after a dilation then point P was the center of dilation.

CONCEPT 2 - Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line passing through the center of dilation unchanged.

Dilation of when the center of dilation is point O on the line with scale factor of k, k > 1. / Dilation of when the center of dilation is point O on the line with scale factor of k, 0 < k < 1.
PRE-IMAGE () / IMAGE (=)
/

The result of dilating by a scale factor of k > 1 wtih the center of dilation that is on the is simply . The dilation maps all points either on to or which are the opposite rays that form when O is between A and B.
/ PRE-IMAGE () / IMAGE (=)
/

The result of dilating by a scale factor of 0 < k < 1 wtih the center of dilation that is on the is simply . The dilation maps all points either on to or which are the opposite rays that form when O is between A and B.

CONCEPT 3 - Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line not passing through the center of the dilation to a parallel line.

PRE-IMAGE () / IMAGE ()
The center of dilation is point O (not on ) and the scale factor is k > 1. / Angles are preserved in dilation,
thus ÐOBA @ ÐOB’A’ and because these
corresponding angles are congruent,
.

It is critical to understand that dilations create parallel lines between
ALL pre-image and image corresponding segments and lines.

CONCEPT 4 - Verify experimentally the properties of dilations given by a center and a scale factor: the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Scale Factor 0 < k < 1 / Scale Factor k > 1
B’C’ BC and
A’C’ AC and
A’B’ AB and / B’C’ > BC and
A’C’ > AC and
A’B’ > AB and

CONCEPT 5 – What happens when the scale factor is a negative number?

A dilation with center O and a scale factor of k is a transformation that maps every point P in the plane to point P’ so that the following properties are true.
1. If P is NOT the center O, then the P’ lies on the OPPOSITE RAY OF when the scale factor k is a negative number such that k = and k ¹ 1.
/ < -1
/ -1 < < 0

Does this look like something
you have already seen?
A Rotation of 180°
/

The negative value rotates
the dilated image 180°.
/
CONCEPT 6 – What happens when the center of dilation is in, on and out of a figure?
Three very distinct diagrams appear for these three situations.
Center of Dilation (Inside) / Center of Dilation (On) / Center of Dilation (Outside)
/ /
/ /

CONCEPT 7 – 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

It is obvious that 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.

Examples using the coordinate rule of dilation when the center of dilation is the origin.

Example #1 A dilation of 2 with center of dilation O, the origin.
Example #2 A dilation of ½ with center of dilation O, the origin.
Example #3 A dilation of -1/3 with center of dilation O, the origin.

Examples using the coordinate rule of dilation when the center of dilation is NOT the origin.

Example #1 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>.
Example #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>.

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