In this unit students will learn about the Distance Formula and its corresponding proof. They will use this formula to find the distance between two points in the Cartesian plane, to prove a quadrilateral is a rectangle (in conjunction with the converse of the Pythagorean theorem), and to determine whether a point whose coordinates are given lies on a circle whose equation is to be found. Students will also learn the definition of a midpoint and the Midpoint Formula, along with its corresponding proof. They will use this formula to find the midpoint between two points in the Cartesian plane and to prove properties of a parallelogram. Moreover, students will use rigid motions to prove theorems using the Distance and the Midpoint formulas in the Cartesian plane.
Students will also investigate and learn the relationship between the slopes of parallel lines in a coordinate plane. The Slopes of Parallel Lines Theorem will be discussed, proved, and used to find the slope of a parallel line to a given line, which may or may not be written in slope-intercept form. The graph of a line and a point not on the line will be given, and students will be asked to find the equation of the parallel line through the point. Systems of equations with parallel or overlapping lines will be discussed, as well as properties of different systems of equations. Properties of parallel lines will be used to classify parallelograms. Furthermore, slope triangles of different lines will be investigated, and students will learn about the relationship between the slopes of perpendicular lines in a coordinate plane. The Slopes of Perpendicular Lines Theorem will be discussed, proved, and used to find the slope of a perpendicular line to a given line, which may or may not be written in slope-intercept form. The equation of a perpendicular line to a given line will be investigated, and a perpendicular line to a line whose graph is given, through a given point, will be found.
This unit will also discuss positions of points in a line segment, the partitioning of a directed line segment, and a general method for partitioning directed line segments. This will be used to find points on a triangle's perimeter and to partition distances in real life examples.
Using the Distance formula and the Pythagorean theorem, students will work with the perimeter and area of different polygons in the Cartesian plane. They will solve real-life problems and investigate properties of polygons and compound shapes.
Finally, the definition of a parabola will be given. Students will investigate points that are equidistant from a pair of given options and properties of parabolas, and will learn about the focus and directrix of a parabola. They will use the Distance Formula to calculate points on a parabola and will write its equation, given the focus and directrix. Vertical and horizontal parabolas will be discussed, as well as the use and applications of parabolas in the real world.