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Objectives: Find distance between two points in the coordinate plane

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.

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Objectives: Find distance between two points in the coordinate plane

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  1. 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

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

  3. I II 0 III IV Midpoint Formula Theorem : If the coordinates of the end points of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint of this segment is given by the formula : Example: Find the midpoint given by the points A(4, 4) and B(-3, -2). A M B Midpoint = Midpoint =

  4. D(-3, 5) A(1, 3) B(5, 3) C(-3, 1) Find the Distance and Midpoint on Vertical and Horizontal Lines 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:___________ 1 - 5= 4 1+ 5, 3 + 3 = (3, 3) midpoint 2 2 5 - 1= 4 -3 + -3, 5 + 1 = (-3, 3) midpoint 2 2

  5. 8 + (–6) 2 2 2 The x–coordinate is = = 1 The midpoint has coordinates Midpoint Formula x1 + x2 2 y1 + y2 2 ( , ) Substitute 8 for x1 and (–6) for x2. Simplify. 6 2 The y–coordinate is = = 3 9 + (–3) 2 Substitute 9 for y1 and (–3) for y2. Simplify. Try It! AB has endpoints (8, 9) and (–6, –3). Find the coordinates of its midpoint M. Use the Midpoint Formula. Let (x1, y1) be A(8, 9) and (x2, y2) be B(–6, –3). The coordinates of midpoint M are (1, 3).

  6. Recall the Pythagorean Theorem c2 = a2 + b2 • a and b are the legs • c is the hypotenuse (the longest length) • only applies to right triangles

  7. Relate Distance Formula to the Pythagorean Theorem Pythagorean Theorem Distance Formula c2 = a2 + b2

  8. 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

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

  10. Real-world and the Distance Formula You are building a fence to enclose an area as shown in the diagram. Approximately, how many feet of fencing will be required? EF = FG = GH = HE = The approximate amount of fencing needed (perimeter) is 5.4 + 5.1 + 5 + 7.1 = 22.6 feet.

  11. 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. 7.2 7.1 (6, -9) (9, - 3) 9.5 Units

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