Distance & Midpoint in the Coordinate Plane

Coordinate Plane x-axis (Independent) y-axis (Dependent) Quad. I ( +, +) Quad. II ( -, +) Quad. III ( -, -) Quad. IV ( +, -) Ordered Pair: (x, y) A(2, -1) A

Graph the following equation. y = 3x methods –x/y chart –Slope-Intercept Form Slope y-intercept

Graphing and finding the distance Graph A(2, -2) & B(2, 6) Graph C(1, 2) & D(5, 6)

Distance Formula Distance Formula: The distance (d) between two points A(x 1, y 1 ) and B(x 2, y 2 ) is d AB = √(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2

Ex.1: Find the distance between the following points. R(-2, 6) & S(4, 4) x1 y1 x2 y2 d RS = √(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = √(4 – (-2)) 2 + (4 – 6) 2 = √(6) 2 + (-2) 2 = √ = √40= 6.3

Find the distance between R(–2, –6) and S(6, –2) to the nearest tenth. Let (x 1, y 1 ) be the point R(–2, –6) and (x 2, y 2 ) be the point S(6, –2). To the nearest tenth, RS = Use a calculator. d = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Use the Distance Formula. d = (6 – (–2)) 2 + (–2 – (–6)) 2 Substitute. d = = d = 82 + (–8) 2 Simplify.

Graphing the Midpoint Graph A(2, -2) & B(2, 6) Graph C(1, 2) & D(5, 6)

MidPoint Formula Midpoint Formula: The coordinates of the midpoint M of AB with endpoints A(x1, y1) and B(x2, y2) are the following: x-coordinate of a point y-coordinate of a point x 2 + x 1 2 y 2 + y 1 2, () M

Midpoint Example AB has endpts A(8, -9) & B(-6, 3). Find the coordinates of its midpt. x 2 + x 1 2 y 2 + y 1 2, M () x1 y1x2 y2 (-6) (-9) 2, M () , M () M(1, -3)

Midpoint Example The midpt of DG is M(-1, 5). One endpt is D(1, 4). Find the coordinates of the other endpts. x 2 + x 1 2 X m = y 2 + y 1 2 y m = x m y m x1 y1 x = (2) -2 = x = x 2 y = (2) 10 = y = y 2 G(-3, 6)

What have we learned??? Distance Formula Midpoint formula d AB = √(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 x 2 + x 1 2 y 2 + y 1 2, () M R(-2, 6) & S(4, 4) x1 y1 x2 y2