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7-2: Exploring Dilations and Similar Polygons

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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. - PowerPoint PPT Presentation

Text of 7-2: Exploring Dilations and Similar Polygons

  • 7-2: Exploring Dilations and Similar PolygonsExpectation: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 ChangesWhen 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 DilationsContraction: 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.

  • DilationsThe following terms all indicate a size change:dilationdilitationcontractionexpansion

  • 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 CoordinatesTo 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 ABC. 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 AB and AB?

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

  • CC is the center of the dilation mapping XYZ onto LMNYXZNML

  • 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 FiguresDefn: 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 FiguresIf WXYZ ~ ABCD, then:W A:X BY C: Z D

  • If ABC is similar to DEF in the diagram below, then mD = ?8060403010

    DFEB80AC40

  • Determine whether the triangles are similar. Justify your response!12953.75139.75

  • Scale FactorThe 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.

  • In the figure below, ABC is similar to DEF. What is the length of DE?A.12B.11C.10D.7E.6

  • Assignmentpages 351-353, #13 25 (odds), 29 and 41

    *7th 2/3/06*