G.SRT.A.2STUDENT NOTES & PRACTICE WS #2–geometrycommoncore.com1

In G.CO.2 we defined a transformation to be a one to one correspondence between the points of the pre-image and the points of the image and then narrowed that definition down to an isometric transformation as a transformation that preserves the distances and angles between the pre-image and image. We will now create another classification of transformations, the similarity transformations.
A similarity transformation is a transformation in which the image has the same shape as the pre-image. Specifically, the similarity transformations are the isometric transformations (reflection, rotation, translation) and dilation as well. The Venn diagram below displays how all these are related to each other.

All transformations are mappings.
All similarity transformations are transformations.
All isometric transformations are similarity transformations. / A Transformation
An Isometric Transformation
Similarity Transformations
REFLECTION
ROTATION
TRANSLATION
DILATION

From this Venn diagram we learn that congruence is a subset of similarity. Congruence requires both same shape and same size whereas similarity only requires same shape.

Definition of Similarity

Two figures are similar if and only if one can be obtained from the other by a single or sequence of similarity transformations.

Notation

If you remember when we introduced the congruence symbol, we presented you with this diagram. The congruence relationship has two facets to it, same measure (equal corresponding lengths) and same shape (equal corresponding angles) thus the symbol includes both parts of that relationship. /
When working with similarity, where only the same shape is required, we see a slight change to the symbol; we only use the ‘squiggle’. /

Writing Similarity Statements

So if we are to write a similarity statement for ABC and its dilated image, DEF. We know these two triangles are similar because the similarity transformation of dilation maps ABC onto DEF. As we did with congruence we correlate the corresponding angles and sides in the name.
/

If ABC DEF then;

ANGLES ARE CONGRUENT / SIDES ARE PROPORTIONAL
A D, B E, & C F
Because similarity transformations
all preserve angles. /
Because DE = AB  k, EF = BC  k & DF = AC  k where k is the scale factor between ABC and DEF, k =
NYTS (Now You Try Some)
1.ABC MRG
M  _____ / ______/ ______
2.STVWQY
T _____ / ______/ ______

Establishing Similarity through Similarity Transformations

Given:
Quadrilateral OBCD
& Quadrilateral OHLK /

O (0,0) --> O (0,0)
B (0,2) --> B’ (0,4)
C (2,3) --> C’ (4,6)
D (3,0) --> D’ (6,0) /

O (0,0) --> O (0,0)
B’ (0,4) --> H (0,-4)
C’ (4,6) --> L (-4,-6)
D’ (6,0) --> K (-6,0)
Given that ABC DEF, we know that a single or sequence of similarity transformations map ABC onto DEF. / First we dilated ABC by the scale factor, so that the two triangles are the same size. This makes the two triangles congruent; they are the same size and the same shape. / Finally there exists a sequence of isometric transformations that map A’B’C’ onto DEF. In this case a rotation of 180 maps A’B’C’ onto DEF.
NYTS (Now You Try Some)
3. Determine the sequence of similarity transformations that map one figure onto the other thus establishing that the two figures are similar.
a) Determine two similarity transformations that would map OBA onto ORG.
______
followed by
______/ b) Determine two similarity transformations that would map Quad. ETHP onto Quad. ODBA.
______
followed by
______