7-2: Exploring Dilations and Similar 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 Changes When we start with one figure and make it bigger or smaller, it is called a size change transformation. The original figure is called the preimage and the resulting figure is called the image.

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

Types of Dilations Contraction: reduction: the image is smaller than the preimage: magnitude is greater than 0, but less than 1. Expansion: enlargement: the image is larger than preimage: magnitude is greater than 1.

Dilations The following terms all indicate a size change: dilation dilitation contraction expansion

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

By what scale factor was the original picture enlarged?

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

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

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

Triangle ABC is a dilation of triangle XYZ. Use the coordinates of the 2 triangles to determine the scale factor of the dilation. A(-1, 1), B(-1, 0), C(3,1) X(-3, 3), Y(-3, 0), Z(9, 3)

Triangle XYZ is a dilation of ΔABC. Use the coordinates of the 2 triangles to determine the scale factor of the dilation. A(-1, 1), B(-1, 0), C(3,1) X(-3, 3), Y(-3, 0), Z(9, 3)

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

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

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

C C is the center of the dilation mapping ΔXYZ onto ΔLMN Y X Z N M L

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

Determine the center of the dilation.

Similar 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 Figures If WXYZ ~ ABCD, then: ∠ W ≅ ∠ A: ∠ X ≅ ∠ B ∠ Y ≅ ∠ C: ∠ Z ≅ ∠ D WX XY YZ WZ AB BC CD AD ===

If ΔABC is similar to ΔDEF in the diagram below, then m ∠ D = ? A. 80° B. 60° C. 40° D. 30° E. 10° D F E B 80° A C 40°

Determine whether the triangles are similar. Justify your response!

Scale 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 y x y - 3 A BC D E F

In the figure below, ΔABC is similar to ΔDEF. What is the length of DE? A.12 B.11 C.10 D.7⅓ E.6⅔ A 10 B 11 C 12 DF8 E

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