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Lesson 9-5 Dilations

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5-Minute Check on Lesson 9-5 Transparency 9-6 Click the mouse button or press the Space Bar to display the answers. Determine whether each regular polygon tessellates the plane. Explain 1.quadrilateral 2.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? Standardized Test Practice: A CB D pentagonquadrilateraltrianglehexagon Yes, Square’s interior angle = 90 360 90 = 4 No, 15-gon’s interior angle = 156 360 156 ≠ integer No, Octagon’s interior angle = 135 360 135 ≠ integer Yes, 3 60 + 2 90 = 360 C No, 108n + 90m ≠ 360

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

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

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Dilations y x 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)

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Dilations y x BA B’ A’ B* A* 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) CP

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

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

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

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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. Draw. Since r is negative, P', Q', R', and S' will lie on respectively. Locate P', Q', R', and S' so that R'R' S'S' P'P' Q'Q'

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

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Example 5-3b COORDINATE GEOMETRY Triangle ABC has vertices A(–1, 1), B(2, –2), and C(–1, –2). Find the image of ABC after 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)

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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

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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: reduction Answer: 2; enlargement

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Example 5-5a MULTIPLE-CHOICE TEST ITEM Sharetta 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 2B 3C D 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.

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

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Example 5-5b Answer: B MULTIPLE-CHOICE TEST ITEM Ruben 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.

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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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