7-2: exploring dilations and similar polygons

7-2: Exploring Dilations and 7-2: Exploring Dilations and Similar PolygonsSimilar Polygons

Expectation:G3.2.1: Know the definition of dilation and find the image of a figure under a dilation.G3.2.2: Given two figures that are images of each other under some dilation, identify the center and magnitude of the dilation.

Size ChangesSize Changes

When we start with one figure and make When we start with one figure and make it bigger or smaller, it is called a size it bigger or smaller, it is called a size change transformation.change transformation.

The original figure is called the The original figure is called the preimagepreimage and the resulting figure is called the and the resulting figure is called the imageimage..

The The magnitude, k,magnitude, k, of a size change is of a size change is how many times bigger (or smaller) the how many times bigger (or smaller) the image is than the preimage.image is than the preimage.

Types of DilationsTypes of Dilations

ContractionContraction: : reductionreduction: the image is : the image is smaller than the preimage: magnitude smaller than the preimage: magnitude is greater than 0, but less than 1.is greater than 0, but less than 1.

ExpansionExpansion: : enlargementenlargement: the image is : the image is larger than preimage: magnitude is larger than preimage: magnitude is greater than 1.greater than 1.

DilationsDilations

The following terms all indicate a size change:

dilation

dilitation

contraction

expansion

A picture is enlarged by a scale factor of A picture is enlarged by a scale factor of 125% and then enlarged again by the 125% and then enlarged again by the same scale factor. If the original picture same scale factor. If the original picture was 4” x 6”, how large is the final was 4” x 6”, how large is the final copy?copy?

By what scale factor was the original By what scale factor was the original picture enlarged?picture enlarged?

Size Changes With Size Changes With CoordinatesCoordinates

To perform a size change of given To perform a size change of given magnitude on a polygon with known magnitude on a polygon with known coordinates, multiply the magnitude of coordinates, multiply the magnitude of the size change by each of the the size change by each of the coordinates of the polygon.coordinates of the polygon.

If D(x,y) = (3x,3y), what is the image of If D(x,y) = (3x,3y), what is the image of the point (-5,8)?the point (-5,8)?

What is the scale factor of the dilation?What is the scale factor of the dilation?

A triangle has coordinates A triangle has coordinates A(3,-1), A(3,-1), B(4,3) and C(2,5). The triangle will B(4,3) and C(2,5). The triangle will undergo a dilation using a scale factor undergo a dilation using a scale factor of 3. Determine the coordinates of the of 3. Determine the coordinates of the vertices of the resulting triangle.vertices of the resulting triangle.

Triangle ABC is a dilation of triangle XYZ. Triangle ABC is a dilation of triangle XYZ. Use the coordinates of the 2 triangles Use the coordinates of the 2 triangles to determine the scale factor of the to determine the scale factor of the dilation. dilation.

A(-1, 1), B(-1, 0), C(3,1)A(-1, 1), B(-1, 0), C(3,1)

X(-3, 3), Y(-3, 0), Z(9, 3)X(-3, 3), Y(-3, 0), Z(9, 3)

Triangle XYZ is a dilation of ΔABC. Use Triangle XYZ is a dilation of ΔABC. Use the coordinates of the 2 triangles to the coordinates of the 2 triangles to determine the scale factor of the determine the scale factor of the dilation. dilation.

A(-1, 1), B(-1, 0), C(3,1)A(-1, 1), B(-1, 0), C(3,1)

X(-3, 3), Y(-3, 0), Z(9, 3)X(-3, 3), Y(-3, 0), Z(9, 3)

Size Change Distance Size Change Distance Theorem Theorem

The image of a segment The image of a segment transformed by a dilation with transformed by a dilation with scale factor k is parallel to and scale factor k is parallel to and |k| times the length of the |k| times the length of the preimage. preimage.

Before a size change, the slope of AB is 4 Before a size change, the slope of AB is 4 and AB = 8. After a size change of and AB = 8. After a size change of magnitude .5, what is the slope of A’B’ magnitude .5, what is the slope of A’B’ and A’B’?and A’B’?

Center of a DilationCenter of a Dilation

The center of any dilation is The center of any dilation is where the lines through all where the lines through all corresponding points intersect.corresponding points intersect.

CC is the center of the dilation mapping ΔXYZ onto ΔLMN

Y

X Z

N

M

L

Given two figures which are dilations of Given two figures which are dilations of each other, how can you find the center each other, how can you find the center of the dilation?of the dilation?

Determine the center of the Determine the center of the dilation.dilation.

Similar FiguresSimilar Figures

Defn: Two figures, F and G, are similar (written F ~ G) iff corresponding angles are congruent and corresponding sides are proportional.

Dilations always result in similar figures!!!

Similar FiguresSimilar Figures

If WXYZ ~ ABCD, then:

∠W ≅ A∠ : X ∠ ≅ B∠

∠Y ≅ C∠ : Z ∠ ≅ D∠

WX XY YZ WZAB BC CD AD= = =

If ΔABC is similar to ΔDEF in the diagram If ΔABC is similar to ΔDEF in the diagram below, then m D = ?∠below, then m D = ?∠

A.A. 80°80°

B.B. 60°60°

C.C. 40°40°

D.D. 30°30°

E.E. 10°10°

D F

E B80°

A C40°

Determine whether the triangles Determine whether the triangles are similar. Justify your response!are similar. Justify your response!

129

5

3.75

13 9.75

Scale FactorScale Factor

The scale factor (magnitude) between similar figures is the ratio of the lengths of corresponding sides.

Triangle ABC is similar to triangle DEF. Determine the scale factor of DEF to ABC (be careful – the order is important), then calculate the lengths of the unknown sides.

12 15

y + 3

9 x

y - 3

A

B C

D

E F

In the figure below, ΔABC is similar to In the figure below, ΔABC is similar to ΔDEF. What is the length of DE?ΔDEF. What is the length of DE?

A.A. 1212

B.B. 1111

C.C. 1010

D.D. 7⅓7⅓

E.E. 6⅔6⅔

A

10

B

11

C12 D F8

E

AssignmentAssignment

pages 351-353, pages 351-353,

#13 – 25 (odds), 29 and 41#13 – 25 (odds), 29 and 41