LAC 28:CXLII.2923. Similarity, Right Triangles, and Trigonometry

A. Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor. a. 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. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Using similarity transformations, determine if two figures are similar using the definition of similarity transformations; and explain the meaning of similarity for triangles as the equality of all corresponding pairs of angles and proportionality of all corresponding sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. B. Prove and apply theorems involving similarity. 1. Prove and apply theorems about triangles. Theorems include but not limited to a line parallel to one side of a triangle divides the other two proportionally, and the converse of this theorem; the Pythagorean Theorem proved using triangle similarity; SAS similarity criteria; SSS similarity criteria; AA similarity criteria. 2. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. C. Define trigonometric ratios and solve problems involving right triangles. 1. Understand that by similarity, side ratios in right triangles, including special right triangles, 30-60-90 and 45-45-90, are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 2. Explain and use the relationship between the sine and cosine of complementary angles. 3. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.