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**Standard(s): **
[MA2015] GEO (9-12) 32 :

[MA2019] GEO-19 (9-12) 32 :

32 ) Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [G-GPE6]

[MA2019] GEO-19 (9-12) 32 :

32. Use coordinates to prove simple geometric theorems algebraically.

Topic D concludes the work of Module 4. In Lesson 12, students find midpoints of segments and points that divide segments into more equal and proportional parts. Students also find locations on a directed line segment between two given points that partition the segment in given ratios (G-GPE.B.6). Lesson 13 requires students to connect this work to proving classical results in geometry (G-GPE.B.4). For instance, the diagonals of a parallelogram bisect one another, and the medians of a triangle meet at the point of the way from the vertex for each. Lesson 14 is an optional lesson that allows students to explore parametric equations and compare them with more familiar linear equations (G-GPE.B.6, G-MG.A.1). Parametric equations make both the variables in an equation dependent on a third variable, usually time. In this lesson, parametric equations model the robot’s horizontal and vertical motion over a period of time. Introducing parametric equations in the Geometry course prepares students for higher-level courses and also represents an opportunity to show coherence between functions, algebra, and coordinate geometry. Students extend their knowledge of parallel and perpendicular lines to lines given parametrically. Students complete the work of this module in Lesson 15 by deriving and applying the distance formula (G-GPE.B.4) and with the challenge of locating the point along a line closest to a given point, again given as a robot challenge.