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Polygon Similarity and Transformations

Determine whether polygons are similar, find scale factor, describe dilations and combinations of transformations. Use transformations to show figures are not similar. Explore properties of dilations.

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Polygon Similarity and Transformations

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  1. 1. Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of EFGH to KLMN. 2.Inthe diagram,DEF ~ HJK.Find the value of x and y. Warm-Up y 6

  2. Activity: p.366 Explore Properties of Dilations • You will need graph paper, protractor/ruler. • Complete #’s 1-5 • Findings:

  3. Lesson 6.2 Transformations & Similarity

  4. EXAMPLE 1 Describe a dilation FEG is similar to FDH. Describe the dilation that moves FEG onto FDH. SOLUTION The figure shows a dilation with center F. The scale factor is 2 because the ratio of FH to FG is 20 : 10, or 2 : 1.

  5. 2 3 EXAMPLE 2 Describe a combination of transformations ABC is similar to FGE. Describe a combination of transformations that moves ABConto FGE. SOLUTION A dilation with center B and scale factor moves ABC onto DBE. Then a rotation of DBE with center E moves DBE onto FGE. The angle of rotation is equal to the measure of C.

  6. 7 2 3 3 for Examples 1 and 2 GUIDED PRACTICE The two figures are similar. Describe the transformation(s) that move the blue figure onto the red figure. 1. 2. ANSWER ANSWER dilation with center B and scale factor dilation with scale factor and reflection

  7. EXAMPLE 3 Use transformations to show figures are not similar Use transformations to explain why ABCDEand KLQRPare not similar.

  8. 2 2 3 3 EXAMPLE 3 Use transformations to show figures are not similar SOLUTION Corresponding sides in the pentagons are proportional with a scale factor of . However, this does not necessarily mean the pentagons are similar. A dilation with center Aand scale factor moves ABCDEonto AFGHJ. Then a reflection moves AFGHJonto KLMNP. KLMNPdoes not exactly coincide with KLQRP, because not all of the corresponding angles are congruent. (Only A and Kare congruent.) Since angle measure is not preserved, the two pentagons are not similar.

  9. How to Construct a Dilation of an Object • Hand-out • Draw an object and select a center of dilation. • The center of dilation can be anywhere and is also referred to as a fixed point.

  10. 1 2 EXAMPLE 4 Use similar figures GRAPHIC DESIGN A design for a party mask is made using all equilateral triangles and a scale factor of . a. Describe transformations that move triangle A onto triangle B. b. Describe why triangles C and D are similar by using the given information.

  11. 1 1 2 2 EXAMPLE 4 Use similar figures SOLUTION • The figure shows a dilation with scale factor • followed by a clockwise rotation of 60°. b. Triangles C and D are similar because all pairs of corresponding sides are proportional with a ratio of and all pairs of corresponding angles of equilateral triangles have the same measure.

  12. for Examples 3 and 4 GUIDED PRACTICE Refer to the floor tile designs shown below. In each design, the red shape is a regular hexagon.

  13. for Examples 3 and 4 GUIDED PRACTICE Tile design 1 is made using two hexagons. Explain why the red and blue hexagons are not similar. 3. 4. Tile design 2 is made using two similar geometric shapes. Describe the transformations that move the blue hexagon to the red hexagon. Tile design 3 shows congruent angles and sides. Explain why the red and blue hexagons are similar, using the given information. 5. 6. If the lengths of all the sides of one polygon are proportional to the lengths of all the corresponding sides of another polygon, must the polygons be similar? Explain.

  14. for Examples 3 and 4 GUIDED PRACTICE SAMPLE ANSWER The red hexagon has all sides congruent, but the blue hexagon has 3 shorter sides and 3 longer sides, so ratios of corresponding side lengths are not constant. 3. 4. dilation followed by a rotation of 30°about the center of the figures All angles are congruent, so angle measure is preserved, and all side lengths are congruent in each hexagon, so the ratio of any two corresponding side lengths is constant. 5. No; even though corresponding sides might be proportional, if corresponding angles are not congruent, the polygons are not similar. 6.

  15. Warm-Up Describe the dilation that moves the smaller figure onto the larger figure. 1. dilation with scale factor 3 and center at the intersection of the thin lines ANSWER

  16. 8 5 Warm-Up Describe the dilation that moves the smaller figure onto the larger figure. 2. dilation with scale factor and center at center of the circles ANSWER

  17. Daily Homework Quiz The coordinates of the vertices of a ABC and its image DEFare given. Describe the transformation(s) that move ABCfigure onto DEF. A(1, 1), B(–2, 2), C(2, 2); D(2, –2), E(–4, –4), F(4, –4) SAMPLE ANSWER dilation with center (0, 0) and scale factor 2, then reflection in the x-axis

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