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Lesson 9-5

Lesson 9-5. Dilations . Transparency 9-6. 5-Minute Check on Lesson 9-5. Determine whether each regular polygon tessellates the plane. Explain quadrilateral octagon 3. 15 -gon

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Lesson 9-5

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  1. Lesson 9-5 Dilations

  2. Transparency 9-6 5-Minute Check on Lesson 9-5 • Determine whether each regular polygon tessellates the plane. Explain • quadrilateral • octagon • 3. 15 -gon • Determine whether a semi-regular tessellation can be created from each figure. Assume each figure has a side length of 1 unit. • 4. triangle and square 5. pentagon and square • 6. Which regular polygon will not tessellate the plane? Yes, Square’s interior angle = 90 360  90 = 4 No, Octagon’s interior angle = 135 360  135 ≠ integer No, 15-gon’s interior angle = 156 360  156 ≠ integer Yes, 360 + 290 = 360 No, 108n + 90m ≠ 360 Standardized Test Practice: A triangle quadrilateral pentagon D hexagon C B C Click the mouse button or press the Space Bar to display the answers.

  3. Objectives • Determine whether a dilation is an enlargement, a reduction, or a congruence transformationenlargement – gotten larger |r| > 1reduction – gotten smaller |r| < 1congruence transformation – stayed the same |r| = 1where r is the scaling factor (similar triangles)

  4. Vocabulary • Dilation – a transformation that may change the size of a figure

  5. y x Dilations Five hexagons that are each 50% reduction (1/2) of the size of the one before it Scale factor is ½, and the center point is the origin (center of the figures)

  6. y x Dilations A B A’ B’ CP A* B* Small dashed lines are the rays along which the dilations occur and the center point is CP . AB = 16, A’B’ = 8 so r = ½ (a reduction |r|<1) A*B* = 8 but since its on the opposite side of CP, r = - ½ (negative = opposite, but still a reduction)

  7. y x Dilations |r| > 1 Enlargement |r| < 1 Reduction |r| = 1 Congruence Transformation 1 < r A B 0 < r < 1 C CP – Center Point CP These are the lines that the figure’s vertices “travel” on asthe figure is dilated C’ -1 < r < 0 A’B’C’ is withr = -1. B’ A’ -1 > r

  8. a. Find the measure of the dilation image or the preimage of using the given scale factor of CD = 15, r = 3. Multiply each side by Example 5-1a Dilation Theorem Answer: 45 Multiply. b. Find the measure of the dilation image or the preimage of using the given scale factor C’D’ = 7, r = - ⅔. Dilation Theorem Answer: 10.5

  9. Find the measure of the dilation image or the preimage of using the given scale factor. a. b. Example 5-1b Answer: 32 Answer: 36

  10. R' S' P' Q' Draw . Since r is negative, P', Q', R', and S' will lie on respectively. Locate P', Q', R', and S' so that Example 5-2a Draw the dilation image of trapezoid PQRS with center C and r = - 3 Since |r| > 1 the dilation is an enlargement of trapezoid PQRS.

  11. Draw the dilation image of trapezoid PQRS with center C and R' Since the dilation is an enlargement of trapezoid PQRS. S' P' Q' Example 5-2a Answer: Draw trapezoid P'Q'R'S'.

  12. Example 5-3b COORDINATE GEOMETRY Triangle ABC has vertices A(–1, 1),B(2, –2), and C(–1, –2). Find the image of ABCafter a dilation centered at the origin with a scale factor of 2. Sketch the preimage and the image. Answer:A'(–2, 2), B'(4, –4), C' (–2, –4)

  13. Example 5-4a Determine the scale factor used for the dilation with center C. Determine whether the dilation is an enlargement, reduction, or congruence transformation. image lengthpreimage length Simplify. Since the image is on the opposite side of C from the preimage, the scale factor is negative. So the scale factor is –1. The absolute value of the scale factor equals 1, so the dilation is a congruence transformation. Answer: –1; congruence transformation

  14. Answer: reduction Example 5-4b Determine the scale factor used for each dilation with center C. Determine whether the dilation is an enlargement, reduction, or congruence transformation. a. b. Answer: 2; enlargement

  15. MULTIPLE-CHOICE TEST ITEMSharetta built a frame for a photograph that is 20 centimeters by 25 centimeters. The frame measures 400 millimeters by 500 millimeters. Which scale factor did she use? A 2 B 3 C D Example 5-5a Read the Test Item The photograph’s dimensions are given in centimeters, and the frame’s dimensions are in millimeters. You need to convert from millimeters to centimeters in the problem.

  16. Example 5-5a Solve the Test Item Step 1 Convert from millimeters to centimeters. or 40 centimeters or 20 centimeters Step 2 Find the scale factor. frame lengthphoto length Simplify. Step 3 Sharetta used a scale factor of 2 to build the frame. Choice A is the correct answer. Answer: A

  17. MULTIPLE-CHOICE TEST ITEMRuben is making a scale drawing of the front of his house. His house is 48 feet wide and 30 feet high at its highest point. Ruben decides on a dilation reduction factor of What size poster board will he need to make a complete drawing? A 19 in. by 26 in. B 22 in. by 30 in. C 20.5 in. by 28 in. D 16 in. by 29 in. Example 5-5b Answer: B

  18. Summary & Homework • Summary: • Dilations can be • enlargements, (|scaling factor| > 1) • reductions, or (|scaling factor| < 1 • congruence transformations (|scaling factor|=1) • Homework: • pg 494-496; 14, 15, 20-23, 30-35

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