Lesson 9-5 Dilations

5-Minute Check on Lesson 9-5 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 B quadrilateral C C pentagon D hexagon Click the mouse button or press the Space Bar to display the answers.

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)

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

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)

Dilations y x 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)

Dilations |r| > 1 Enlargement 1 < r |r| < 1 Reduction A y x |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 as the figure is dilated C’ -1 < r < 0 A’B’C’ is with r = -1. B’ A’ -1 > r

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

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

Draw the dilation image of trapezoid PQRS with center C and r = - 3 Since |r| > 1 the dilation is an enlargement of trapezoid PQRS. 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 Since the dilation is an enlargement of trapezoid PQRS. R' S' P' Q' Answer: Draw trapezoid P'Q'R'S'. Example 5-2a

Answer: A'(–2, 2), B'(4, –4), C' (–2, –4) 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) Example 5-3b

image length preimage length 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 Example 5-4a

Determine the scale factor used for each dilation with center C 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 Example 5-4b

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 2 B 3 C 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. Example 5-5a

Step 1 Convert from millimeters to centimeters. 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 Example 5-5a

factor of What size poster board will he need to 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. Answer: B Example 5-5b

Summary & Homework Summary: Homework: 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