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Dilation Similarity Theorem: If one polygon is a dilated image of another polygon, then the polygons are similar. A dilation is a type of transformation that changes the size of an image based on a central point and a scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Dilation means choosing a point to be the center, and drawing lines that go through the center and a vertex of the object. To make an object that is similar to the original object, you can draw lines that are parallel to the lines of the object. The intersections with the lines that go through the center are then the points of the new similar object. Scale Factor=1, the figures are congruent. Scale factor between 0 and 1, the image is smaller. Scale factor > 1, the image is bigger. Dilating a line that does not pass through the center of dilation d. What is the relation between the original line and its dilated image ? Solution: d. When dilating, a line that does not pass through the center of dilation, is parallel to the original line. Example Y’(6,15) Y(2,5) R(2,2) R’(6,6) O(6,2) O’(18,6) Where (a,b) are the x and y coordinates of the center of dilation; and k is the scale factor Of the dilation. As a result, when the center of dilation is the origin (0,0), then: 𝑫 𝟎,𝟎 ,𝒌 𝑫 𝒙, 𝒚 = (𝟎 + 𝒌 𝒙 − 𝟎 , 𝟎 + 𝒌(𝒚 − 𝟎) 𝟎,𝟎 ,𝒌 𝒙, 𝒚 = (𝒌𝒙, 𝒌𝒚) 11.2: Similar Triangles: 3 Similarity shortcuts 1. AA Similarity shortcut: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example: Are triangles A & B similar? By the triangle sum theorem 57°+ 83° + x = 180° X= 40° x A B Yes, By AA similarity theorem 2. SSS Similarity shortcut: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar. Example: Are Triangles # 1 and # 2 similar? Yes, 𝟔 𝟖 𝟏𝟎 #1 #2 = = 𝟑 𝟒 By SSS similarity theorem 𝟓 3. SAS Similarity shortcut: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar. Example: Are the triangles below similar? 𝟏𝟐 = 𝟏𝟓 𝟑𝟔 𝟒𝟓 Yes, they are similar, by SAS similarity theorem In summary, there are 3 similarity shortcuts to prove that two triangles are similar: 1. AA similarity shortcut 2. SSS similarity shortcut 3. SAS similarity shortcut Homework Workbook 11.2 Except # 8 Classwork Work with a partner Hint: Use Pythagorean Theorem a² + b² = c² Hint: Use proportions, then use the Pythagorean Theorem a² + b² = c² No, not all isosceles triangles are similar No, not all right triangles are similar Yes. All isosceles right triangles are similar