Section 1-6 The Coordinate Plane SPI 21E: determine the distance and midpoint when given the coordinates of two points Objectives: Find distance between two points in the coordinate plane Find the coordinates of the midpoint of a segment 10 miles

0 The Coordinate Plane Vocabulary Origin x-axis(real number line) X y-axis(imaginary number line) Ordered Pair (x, y) coordinates (1, 3) Quadrants: Quadrant I (+, +) (-, +) Quadrant II Quadrant III Quadrant IV (+, -) (-, -) Y

Midpoint Formula Theorem : If the coordinates of the end points of a segment are (x 1, y 1 ) and (x 2, y 2 ), then the coordinates of the midpoint of this segment is given by the formula : 0 Example: Find the midpoint given by the points A(4, 4) and B(-3, -2). A B Midpoint = M

Find the Distance and Midpoint on Vertical and Horizontal Lines A(1, 3) B(5, 3) C(-3, 1) D(-3, 5) Find the horizontal length of AB: Subtract x coordinates: _______________ Find the midpoint of AB: Find the vertical length of CD: Subtract y coordinates: _______________ Find the midpoint of CD:___________   = , = (3, 3) midpoint 2 2   = , = (-3, 3) midpoint 2 2

Use the Midpoint Formula. Let (x 1, y 1 ) be A(8, 9) and (x 2, y 2 ) be B(–6, –3). The coordinates of midpoint M are (1, 3). AB has endpoints (8, 9) and (–6, –3). Find the coordinates of its midpoint M. The midpoint has coordinatesMidpoint Formula (, ) x 1 + x 2 2 y 1 + y 2 2 Substitute 8 for x 1 and (–6) for x 2. Simplify. 8 + (–6) 2 The x–coordinate is = = Substitute 9 for y 1 and (–3) for y 2. Simplify. 9 + (–3) 2 The y–coordinate is = =

c 2 = a 2 + b 2 a and b are the legs c is the hypotenuse (the longest length) only applies to right triangles

Relate Distance Formula to the Pythagorean Theorem Pythagorean Theorem c 2 = a 2 + b 2 Distance Formula

Use the Distance Formula to find the distance between points F and G, to the nearest tenth. Write the distance formula. Substitute in known values. Simplify the Equation

Find the distance between R(–2, –6) and S(6, –2) to the nearest tenth. Using the Distance Formula Use the distance formula Substitute Simplify

You are building a fence to enclose an area as shown in the diagram. Approximately, how many feet of fencing will be required? FG = GH =HE = The approximate amount of fencing needed (perimeter) is = 22.6 feet. Real-world and the Distance Formula EF =

DO NOW Classwork: The Distance and Midpoint Formulas 1.Find the distance between the endpoints M(2, –1) and N(–4, 3) to the nearest tenth. 2.Find the distance between P(–2.5, 3.5) and R(–7.5, 8.5) to the nearest tenth. 3.Find the coordinates of AB, given the endpoint A(2, -3) and the midpoint is M(4, - 6). 4.Find the midpoint of CD, C(6, –4) and D(12, –2). 5.Find the perimeter of triangle RST to the nearest tenth of a unit (6, -9) (9, - 3) 9.5 Units