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Subject Geometry Academic Grade Montclair Public Schools CCSS Geometry Academic Unit: Marshall A.b. 10 Unit # 3 Pacing 8-10 weeks Similarity, Trigonometry, & Transformations Unit Name Overview Unit 3 takes previously learned geometric figures and transforms them (rotations, reflections, translations, dilations) throughout the coordinate plane. Students should be familiar recognizing the transformations as well as perform indicated transformations. Based on knowledge of dilations and other transformations, students will begin to understand similarity of figures. Students will reuse previous information from congruence in Unit 2 to develop similarity statements. Students will use the idea of similarity to explore what happens in the case of a right triangle and be able to find geometric mean. Right triangles will be studied in depth. Pythagorean Theorem, trigonometric ratios, Law of Sines/Cosines and finding area of non-right triangles will be explored. Standard # Standard MC, SLO Student Learning Objectives Depth of SC, # Knowledge or AC Prove theorems about right triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity G.SRT.4 1 2 3 4 5 6 MC 7 8 9 Apply the Side, Side, Side Similarity Theorem Apply the Side, Angle, Side Similarity Theorem Apply the Triangle Proportionality Theorem and its converse Apply the Side Splitter Theorem Apply the Angle Bisector Theorem Apply the fact that similar triangles have corresponding medians, altitudes, and angle bisectors that are proportional Apply the fact that similar triangles have proportional perimeters Apply the fact that when an altitude is drawn from the right angle to the hypotenuse, the two triangles formed are similar to the given triangle, and to each other Apply the fact that the altitude drawn to the hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse 4 4 4 4 4 4 4 4 4 4 10 11 12 13 G.SRT.6 G.SRT.7 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Explain and use the relationship between the sine and cosine of complementary angles. 14 MC 15 MC 16 G.SRT.8 G.SRT.9 G.SRT.10 G.SRT.11 G.SRT.1 MC Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side Prove the Laws of Sines and Cosines and use them to solve problems Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces) 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 17 18 19 20 SC SC 21 22 23 24 MC 25 SC 26 Apply the fact that each leg of the original right triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to it Apply the Pythagorean Theorem and its converse Apply the relationship between the sides of a 45-45-90 triangle Apply the relationships between the sides of 30-60-90 right triangles Apply the concepts of similar triangles and scale factors to derive the Sine, Cosine and Tangent functions in terms of acute angles of a right triangle 4 4 4 4 Prove, using the definitions of the three basic trigonometric ratios, that the Sine of one of the acute angles in a right triangle equals the Cosine of the other acute angle. Construct and solve appropriate trigonometric ratios in right triangles to find specific side lengths or angle measures Differentiate angles of elevation from angles of depression Construct right triangles and label them appropriately, to represent relationships in a descriptive scenario Model and solve a series of descriptive scenarios, using right triangle models. Apply right triangle trigonometry to derive the equation A = 1/2 ab sin(C) from the equation A=1/2bh 4 Apply the Law of Sines to solve triangles Apply the Law of Cosines to solve triangles Apply a combination of the Law of Sines and Law of Cosines to solve real world problems Apply the Law of Sines and Law of Cosines to solve a range of real world problems 4 4 4 Apply the properties of dilations to show that a dilation takes a line not passing through the center to a parallel line Apply the properties of dilations to show that a dilation leaves a line passing through the center unchanged 4 4 4 4 3 4 4 G.SRT.2 G.SRT.3 G.CO.2 G.CO.3 G.CO.4 G.CO.5 G.CO.6 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. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch) Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent MC 27 Apply the properties of dilations to determine under what conditions a line segment will become longer or shorter 28 Apply the definition of similarity in terms of similarity transformations to decide if two figures are similar Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 4 30 Apply the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 4 31 Develop and perform rigid transformations that include reflections, rotations, translations and dilations using geometric software, graph paper, tracing paper, and geometric tools and compare them to non-rigid transformations. 4 32 Investigate which transformations (rotations and reflections) will carry the given polygon onto itself. 3 33 Apply the definitions of angles, circles, parallel lines, perpendicular lines and line segments to describe rotations, reflections, and translations. 4 34 Apply the properties of transformations to draw a given geometric figure and its images using graph paper, tracing paper or geometric software. Identify a sequence of transformations that will carry a given figure onto another. 4 Apply rigid transformations to determine, explain and prove congruence of geometric figures 4 29 MC MC MC MC MC 35 36 MC 4 2 Big Ideas Right triangles Pythagorean Theorem Right Triangle Trigonometry Transformations of geometric shapes Law of Sines and Cosines Area of a triangle with Trigonometry Essential Questions How can the Pythagorean Theorem be used to classify a triangle? What happens to a geometric figure under a reflection, translation, rotation, dilation and a combination of any of these? How can trigonometry be used to find the area of a triangle? How can right triangle trigonometry be used to solve right triangles? How does the Pythagorean Theorem and Right Triangle Trigonometry apply to real world problems? Key Vocabulary Sine, Cosine and Tangent Functions Rotation, reflection, translation and dilation Solving a triangle Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards) Chapter 9 Chapter 12