Transcript Document
Lesson 9-5 Dilations Transparency 9-6 5-Minute Check on Lesson 9-5 Determine whether each regular polygon tessellates the plane. Explain 1. quadrilateral Yes, Square’s interior angle = 90 2. octagon No, Octagon’s interior angle = 135 3. 15 -gon No, 15-gon’s interior angle = 156 360 90 = 4 360 135 ≠ integer 360 156 ≠ integer 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 No, 108n + 90m ≠ 360 Yes, 360 + 290 = 360 6. Standardized Test Practice: Which regular polygon will not tessellate the plane? A triangle B quadrilateral C pentagon Click the mouse button or press the Space Bar to display the answers. D hexagon 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 A B A’ B’ x 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 y |r| > 1 Enlargement 1<r A |r| < 1 Reduction B |r| = 1 Congruence Transformation 0<r<1 C CP These are the lines that the figure’s vertices “travel” on as the figure is dilated C’ -1 < r < 0 B’ A’ -1 > r x CP – Center Point A’B’C’ is with r = -1. 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 Find the measure of the dilation image or the preimage of using the given scale factor. a. Answer: 32 b. Answer: 36 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 and S' will lie on . Since r is negative, P', Q', R', respectively. Locate P', Q', R', and S' so that Draw the dilation image of trapezoid PQRS with center C and Since the dilation is an enlargement of trapezoid R' PQRS. S' P' Q' Answer: Draw trapezoid P'Q'R'S'. 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) 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 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 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. 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 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 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