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Geometry 2: Triangle Similarity Part 1 REVIEW Key G-CO.2. Learning Target: I can compare transformations that preserve distance and angle measures to those that do not. 1. Compare the two transformations shown below, with respect to distance and angle measures. Transformation (a): In (b), the transformation stretches the figure only in the horizontal direction and does not preserve the angle measurements, thus the figures are not similar. G.CO-10. Learning Target: I can prove that the segment joining midpoints of two sides of a triangle is parallel to and half the length of the third side. 2. The coordinates of the vertices of a triangle are K(2,3), L(-2,-1), and M(5,1). (a) Find the coordinates of N, the midpoint of KM , and P, the midpoint of LM . 2+5 3+1 , 2 ) 2 In (a), the transformation (translation) is a rigid motion so the lengths of all corresponding sides and the size of all corresponding angles are congruent. Transformation (b): β2+5 β1+1 , 2 ) 2 π: ( π: ( π: (3.5,2) π: (1.5,0) (b) Show that NP || KL . πππ Μ Μ Μ Μ = 2β0 3.5β1.5 = =1 2 2 3+1 4 πΜ Μ Μ Μ πΎπΏ = 2+2 = 4 = 1 (c) Show that NP = ½ KL. 2 2 πΜ Μ Μ Μ ππ = β(3.5 β 1.5) + (2 β 0) = β8 = 2β2 2 2 πΜ Μ Μ Μ πΎπΏ = β(2 + 2) + (3 + 1) = β32 = 4β2 G-SRT.1. Learning Target: I can verify the following statements by making multiple examples: a dilation of a line is parallel to the original line if the center of dilation is not on the line; a dilation of a line segment changes the length by a ratio given by the scale factor. 3. (a) Graph DE with D(-3,-4) and E(4,-4). Then graph its dilation using the origin as the center and a scale factor of 1.5. Label the dilation D ' E ' . G-SRT.2.Learning Target: I can decide if two figures are similar based on similarity transformations. I can use similarity transformations to explain the meaning of similar triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 4. Su, who is 5 feet tall, is standing at point D in the drawing below. The top of her head is at point E. A tree in the yard is at point B with the top of the tree at point C. Su stands so her shadow meets the end of the tree's shadow at point A. Part A: Which triangles are similar? Part B: How do you know? The triangles are similar because all corresponding angles are congruent. π« Part C: What is the ratio of the triangles? π¬ π«β² (βπ. π, βπ) π¬β² (π, βπ) 5 8 Part D: Find the height of the tree. 15 ft 5.Find the value of x, y, and the measure of angle P. (b) Find the length of the dilated line segment. π· β² πΈ β² = ππ. π (c) Find the value of the ratio of the dilated segment to the length of the original segment. 10.5 1.5 = 7 1 The ratio of the length of the dilated segment to the length of the original segment is 1.5: 1. 21 2 xο½6 yο½ P ο½ 860 6. Use side length proportionality to determine if the triangles are similar. π π π = = π. π ππ. π ππ π π π = = π. π π. π π. π The triangles are similar G-SRT.3. Learning Target: I can establish the AA criterion by looking at multiple examples using similarity transformation of triangles. 7. Use the Angle-Angle Similarity Postulate to determine which pair of triangles is not similar. A) . C). B). D).