UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 6: Defining and Applying Similarity Instruction

Transcription

1 Prerequisite Skills This lesson requires the use of the following skills: creating ratios solving proportions identifying congruent triangles calculating the lengths of triangle sides using the distance formula recognizing transformations performed as a combination of translations, reflections, rotations, and/or dilations Introduction Congruent triangles have corresponding parts with angle measures that are the same and side lengths that are the same. If two triangles are congruent, they are also similar. Similar triangles have the same shape, but may be different in size. It is possible for two triangles to be similar but not congruent. Just like with determining congruency, it is possible to determine similarity based on the angle measures and lengths of the sides of the triangles. Key Concepts To determine whether two triangles are similar, observe the angle measures and the side lengths of the triangles. When a triangle is transformed by a similarity transformation (a rigid motion [reflection, translation, or rotation] followed by a dilation), the result is a triangle with a different position and size, but the same shape. If two triangles are similar, then their corresponding angles are congruent and the measures of their corresponding sides are proportional, or have a constant ratio. The ratio of corresponding sides is known as the ratio of similitude. The scale factor of the dilation is equal to the ratio of similitude. Similar triangles with a scale factor of 1 are congruent triangles. Like with congruent triangles, corresponding angles and sides can be determined by the order of the letters. If ABC is similar to DEF, the vertices of the two triangles correspond in the same order as they are named. U1-37

2 The symbol shows that parts are corresponding. A D; they are equivalent. B E ; they are equivalent. C F ; they are equivalent. The corresponding angles are used to name the corresponding sides. AB DE BC EF AC DF AB BC AC = = DE EF DF Observe the diagrams of ABC and DEF. The symbol for similarity ( ) is used to show that figures are similar. ABC DEF B A D B E C F AB BC AC = = DE EF DF A C D F E Common Errors/Misconceptions incorrectly identifying corresponding parts of triangles assuming corresponding parts indicate congruent parts assuming alphabetical order indicates congruence changing the order of named triangles, causing parts to be incorrectly interpreted as congruent U1-38

3 Guided Practice Example 1 Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer Y B X A C -4-5 Z Examine the orientation of the triangles. ABC XYZ Side Orientation Side Orientation AB Top left side of triangle XY Top left side of triangle AC Bottom left side of triangle XZ Bottom left side of triangle BC Right side of triangle YZ Right side of triangle The orientation of the triangles has remained the same, indicating translation, dilation, stretch, or compression. U1-39

4 . Determine whether a dilation has taken place by calculating the scale factor. First, identify the vertices of each triangle. A ( 4, 1), B (1, 4), and C (, ) X ( 8, ), Y (, 8), and Z (4, 4) Then, find the length of each side of ABC and XYZ using the distance formula, d= ( x x ) + ( y y ). 1 Calculate the distance of AB d = [(1) ( 4)] + [(4) (1)] d = (5) + (3) d = 5+ 9 Substitute ( 4, 1) and (1, 4) ) and (x, y d = 34 The distance of AB is 34 units. Calculate the distance of BC. 1 1 d = [() (1)] + [( ) (4)] d = (1) + ( 6) d = Substitute (1, 4) and (, ) ) and (x, y d = 37 The distance of BC is 37 units. (continued) U1-330

5 Calculate the distance of AC. 1 1 d = [() ( 4)] + [( ) (1)] d = (6) + ( 3) d = d = 45 = 9 5 = 3 5 The distance of AC is 3 5 units. Substitute ( 4, 1) and (, ) ) and (x, y Calculate the distance of XY. 1 1 d = [() ( 8)] + [(8) ()] d = (10) + (6) d = d = 136 = 4 34 = 34 Substitute ( 8, ) and (, 8) ) and (x, y The distance of XY is 34 units. (continued) U1-331

6 Calculate the distance of YZ. 1 1 d = [(4) ()] + [( 4) (8)] d = () + ( 1) d = d = 148 = 4 37 = 37 The distance of YZ is 37 units. Substitute (, 8) and (4, 4) ) and (x, y Calculate the distance of XZ. 1 1 d = [(4) ( 8)] + [( 4) ()] d = (1) + ( 6) d = d = 180 = 36 5 = 6 5 The distance of XZ is 6 5 units. Substitute ( 8, ) and (4, 4) ) and (x, y U1-33

7 3. Calculate the scale factor of the changes in the side lengths. Divide the side lengths of XY 34 = = AB 34 YZ 37 = = BC 37 XZ 6 5 = = AC 3 5 XYZ by the side lengths of ABC. The scale factor is constant between each pair of corresponding sides. 4. Determine if another transformation has taken place. Multiply the coordinate of each vertex of the preimage by the scale factor, k. D k (x, y) = (kx, ky) D ( 4, 1) = [( 4), (1)] = ( 8, ) D (1, 4) = [(1), (4)] = (, 8) D (, ) = [(), ( )] = (4, 4) You can map Dilation by a scale factor of k ABC onto XYZ by the dilation with a scale factor of. 5. State your conclusion. A dilation is a similarity transformation; therefore, ABC and XYZ are similar. The ratio of similitude is. U1-333

8 Example Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer P J N H K Q Examine the angle measures of the triangles. Use a protractor or construction methods to determine if corresponding angles are congruent. None of the angles of HJK are congruent to the angles of NPQ.. Summarize your findings. Similarity transformations preserve angle measure. The angles of HJK and NPQ are not congruent. There are no sequences of transformations that will map HJK onto NPQ. HJK and NPQ are not similar triangles. U1-334

9 Example 3 A dilation of TUV centered at point P with a scale factor of is then reflected over the line l. Determine if TUV is similar to DEF. If possible, find the unknown angle measures and lengths in DEF. 5 V D T 3 U P E l F 1. Determine if TUV and DEF are similar. The transformations performed on TUV are dilation and reflection. The sequence of dilating and reflecting a figure is a similarity transformation; therefore, TUV DEF. U1-335

10 . Identify the angle measures of DEF. Corresponding angles of similar triangles are congruent. T D U E V F Since U is marked as a right angle, E must also be a right angle. 3. Identify the known lengths of DEF. Corresponding sides of similar triangles are proportional. The ratio of similitude is equal to the scale factor used in the dilation of the figure. DE EF DF = = = TU UV TV Since the length of TU is 3 units, the length of the corresponding side DE can be found using the ratio of similitude. DE = 3 DE = 6 DE is 6 units long. Since the length of TV is 5 units, the length of the corresponding side DF can be found. DF = 5 DF = 10 DF is 10 units long. U1-336