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Dilations and Similarity in the Coordinate Plane A dilation is a transformation that changes the size of a figure but not its shape. The preimage and image are always similar. A scale factor describes how much a figure is enlarged or reduced. ABC Triangle ABC has vertices A(0, 0), B(2, 6), and ABC C(6, 4). Find the coordinates of the vertices of 1 the image after a dilation with a scale factor . 2 Preimage Image ABC ABC 1 1 A(0, 0) 0 , 0 A(0, 0) 2 2 1 1 B(2, 6) 2 , 6 B(1, 3) 2 2 1 1 C(6, 4) 6 , 4 C(3, 2) 2 2 FEG HEJ. Find the coordinates of F and the scale factor. FE EG Write a proportion. HE EJ FE 4 HE 6, EG 4, and EJ 8. 6 8 8(FE) 24 Cross Products Property FE 3 Divide both sides by 8. So the coordinates of F are (0, 3). Since F(0, 3) (0 • 2, 3 • 2) H(0, 6), the 2 scale factor is . 1 Dilations and Similarity in the Coordinate Plane You can prove that triangles in the coordinate plane are similar by using the Distance Formula to find the side lengths. Then apply SSS Similarity or SAS Similarity. Use the figure to prove that ABC ADE. Step 1 Determine a plan for proving the triangles similar. AB A A by the Reflexive Property. If AD AC , then the triangles are similar by SAS . AE Step 2 Use the Distance Formula to find the side lengths. AB 1 3 2 4 1 2 AC 13 AD Step 3 1 3 5 3 2 3 1 2 8 2 2 2 7 1 2 AE 7 3 2 5 1 2 52 2 13 32 4 2 Compare the corresponding sides to determine whether they are proportional. AB 13 1 AD 2 13 2 AC 2 2 1 AE 4 2 2 The similarity ratio is 1 AB AC , and . So ABC ADE by SAS . 2 AD AE 5. Prove that FGH FLM. ________________________________________ ________________________________________ 6. Prove that QRS TUV. ________________________________________ Name _______________________________________ Date __________________ Class __________________ Triangle Similarity: AA, SSS, and SAS Angle-Angle (AA) Similarity Side-Side-Side (SSS) Similarity Side-Angle-Side (SAS) Similarity If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. ABC DEF ABC DEF ABC DEF Explain how you know the triangles are similar, and write a similarity statement. 1. 2. ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ 3. Verify that ABC MNP. ________________________________________ ________________________________________ Name _______________________________________ Date __________________ Class __________________ Choose the best answer. 6. Complete the similarity statement. 1. Two points on a line are M(4, 5) and N(2, 9). Which ratio represents the slope of the line? A B 2 3 1 7 C 2 3 D 7 2. The ratio of the angle measures of a pentagon is 4 : 5 : 3 : 8 : 7. What is the measure of the smallest angle? F 27° H 60° G 40° J 81° 3. Three sides of a triangle measure 0.5, 0.6, and 0.8. Two sides of a similar triangle measure 2 and 3.2. What is the length of the third side? A 0.06 C 1.28 B 0.8 D 2.4 4. An Eiffel Tower mural is 2 meters high and 0.77 meters wide. If the actual tower is 125 meters wide, how tall is it to the nearest meter? F 48 m H 325 m G 125 m J 842 m ABC ? F ZYX H XZY G YZX J XYZ 7. Rob is making for his young son a replica armchair of one he uses. The ratio of the length of the arm of the old chair to that of the new chair is 3 : 2. The arm of the old chair is 9 inches long. How many inches long will the arm of the new chair be? A 6 C 13 B 8 D 18 1 2 8. Which two triangles could you prove similar by AA ? F H G J 5. Which is similar to this quadrilateral? 9. To measure the distance across a pond, a surveyor locates points A, B, C, D, and E as shown. What is AB to the nearest meter? A C B D A 10 C 15 B 12 D 18 Name _______________________________________ Date __________________ Class __________________ Answers for Multiple qustions 1. A 6. F 2. H 7. A 3. D 8. F 4. H 9. C 5. B