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Warm up Solve each equation: . x 2x 5 4 . 3x 4 5 Lesson 10-1 Introduction to Analytical Geometry Objective: To find the distance and midpoint between two points on a coordinate plane To prove geometric relationships among points and lines using analytical methods Analytical Geometry The study of coordinate geometry from an algebraic perspective. Distance on a Number Line Distance between 2 points a and b on a number line = a b or b a Distance Formula If d is the distance between two points with coordinates (x1, y1) and (x2, y2) then d x2 x1 y2 y1 2 2 Distance Formula d x2 x1 2 y2 y1 2 Where d stands for distance x1 & y1 are one endpoint of a segment x2 & y2 are the second endpoint of a segment (x2 , y2) (x1 , y1) d Example Find the distance between points at (4, -2) and (8, 3). Midpoint Formula for a Coordinate Plane On a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1,y1) and (x2,y2) are x1 + x2, y1 + y2 2 2 Midpoint (x1, y1) x1 + x2, y1 + y2 2 2 (x2, y2) Practice Find the coordinates of the midpoint of the segment that has endpoints at (2, 5) and (-4, -7) Practice Determine whether quadrilateral PQRS with vertices P(-4, 2), Q(-3, -2), R(3, -3) and S(1, 5) is a parallelogram. Practice Prove that the diagonals of a square are perpendicular bisectors of each other. D(0, a) C(a, a) A(0, 0) B(a, 0)