Lesson 1: Introductions to Dilations
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1 : Introductions to Dilations Learning Target I can create scale drawings of polygonal figures I can write scale factor as a ratio of two sides and determine its numerical value A dilation is a transformation whose preimage and image are similar. Thus, a dilation is a similarity transformation. It is not, in general, a rigid motion. Every dilation has a center and a scale factor n, n > 0. The scale factor describes the size change from the original figure to the image. A dilation with center R and scale factor n, n > 0, is a transformation with the following properties: A dilation that creates a larger image is called an enlargement. When the image is an enlargement the scale factor is greater than 1. The image expands. 2 Types of Dilations A dilation that creates a smaller image is called a reduction. When the image is reduced that the scale factor is between 0 and 1. The image contracts.
2 The Notation for Dilation is: D O,r where O: of and r: Definition: A dilation is a rule (a function) that moves points in the plane a specific distance along the ray that originates from a center O. What determines the distance a given point moves? 1. If r > 1 the dilation will push the point from the center. 2. If r = 1 the dilation will keep the point from the center of dilation 3. If 0 < r < 1 the dilation will pull the point the center. Finding the Scale Factor Example 1. Find the scale factor The ratio of corresponding sides is the scale factor (n) of the dilation. Example 2. The dashed line figure is a dilation image of the solid-line figure. D is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. Example 3. The image of EDF after a dilation of scale factor k centered at point D is GDH, as shown in the diagram below. What is the scale factor? ( give as a ratio of two sides) Quick Write Is a dilation a rigid motion? Explain
3 Drawing Dilation Images Example 3. Draw the dilation image ΔB C D = D (2,X) (ΔBCD) Example 4. Given center O and triangle ABC, dilate the figure from center O by a scale factor of r = 1 4. Label the dilated triangle A B C Connect the center of dilation O to all vertices Measure the length of segments OA =, OB=, OC= Apply the dilation rule for each length and calculate OA =, OB =, OC = Remember your dilated points A,B,C will be on the same rays that connect the center of dilation to each vertices Measure each length and connect the new points, label the image A B C Find the following ratios OC OC = OB OB OA = = OA Write the scale factor as a ratio of sides
4 Example 4 (continued) A line segment AB undergoes a dilation of r = 1 4. What will the length of the image segment A B? Angle CBA measures 78. After a dilation, what will the measure of C B A be? How do you know? Example 5 Given center O and triangle ABC, dilate the triangle from center O with a scale factor r = 3. Measure the length of segments OA =, OB=, OC= Apply the dilation rule for each length and calculate OA =, OB =, OC = Find the following ratios OC OC = OB OB OA = = OA Set up an extended proportion of the corresponding lengths Using a ruler measure AB= BC = and AC = = = Using a ruler measure A B = B C = A C = Set up an extended proportion of the corresponding side-lengths = = = Lesson Summary There are two properties of a scale drawing of a figure: 1. Corresponding angles are equal in measurement 2. Corresponding lengths, sides are proportional in measurement.
5 : Introductions to Dilations Classwork Exercise 1. The solid-line figure is a dilation of the dashed-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation a. b. Exercise 2: Find the scale factor As a ratio of sides Numerical Value Exercise 3. The image of after a dilation of scale factor k centered at point A is, as shown in the diagram below. What is the scale factor? Exercise 4. In the diagram below, is the image of after a dilation of scale factor k with center E. Which ratio is equal to the scale factor k of the dilation?
6 Exercise 5. A triangle is dilated by a scale factor of 3 with the center of dilation at the origin. Which statement is true? 1) The area of the image is nine times the area of the original triangle. 2) The perimeter of the image is nine times the perimeter of the original triangle. 3) The slope of any side of the image is three times the slope of the corresponding side of the original triangle. 4) The measure of each angle in the image is three times the measure of the corresponding angle of the original triangle. Exercise 6. In the diagram below, is the image of after a dilation centered at the origin. What is the scale factor? As a ratio of sides Numerical Value Exercise 7) Create a scale drawing/ dilation of the figure below about center O with scale factor r = 1 2.
7 : Introductions to Dilations Homework 1. Create a scale drawing of the figure below about center O and scale factor r = 3. Measure the length ofoa : Measure the length of OA ' : What is the ratio of OA to OA? Write the scale factor as a ratio of two sides m A = 17, m B = 134, m C = 22, m D = 23 What will be the measures of m A = m B =, m C =, m D =
8 2. Dilate circle A, from center O at the origin by scale factor r = Use the ratio method to create a scale drawing about center O with a scale factor of r = 1 4. Give the proper notation of the Dilation:
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LEVEL 9 Mathematics Observation Student: Assessment Date: Grade in School: Concepts Evaluated Score Notes. Applying the concept of slope to determine rate of change Equation of a line: slope-intercept