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Graphs

Explore Rectangular Coordinate Plane, Distance & Midpoint Formulas. Graph ordered pairs, determine quadrants, calculate distances, verify triangle properties, and solve problems using Pythagorean Theorem on a Cartesian plane. Step-by-step explanations included.

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Graphs

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  1. Graphs Rectangular Coordinates Use the distance formula. Use the midpoint formula.

  2. Graphing ordered pairs. • x axis: horizontal line • y axis: vertical line • Origin: point of intersection of the two axes • Rectangular (Cartesian) Coordinate Plane: plane formed by the x-axis and the y-axis • Divided into four sections called quadrants. 1st is upper right, 2nd is upper left, 3rd is lower left, 4th is lower right

  3. Ordered Pair: (x, y) • First value is the x coordinate (abscissa): tells right and left movement • Second value is the y coordinate (ordinate): tells up and down movement

  4. Graph the following ordered pairs and state what quadrant they are in • (-2, 5) Left 2, up 5: 2nd quadrant • (-7, -2.5) Left 7, down 2.5 (approximate the .5): 3rd quadrant • (0,8) No motion right and left, up 8: not in a quadrant • (3, -6) Right 3, down 6: 4th quadrant

  5. Distance Formula • (1, 3) and (5, 6) Find horizontal distance by subtracting x’s. 5 – 1 = 4 Find vertical distance by subtracting y’s. 6 – 3 = 3 Pythagorean Theorem a2 + b2 = c2 42 + 32 = c2 16 + 9 = c2 25 = c2 5 = c

  6. Distance Formula • D = sqrt ((x2 – x1)2 + (y2 – y1)2)) • Remember: This is an application of the Pythagorean Theorem • Find the distance between (-4,5) and (3,2) • Sqrt ((-4 – 3)2 + (5 – 2)2) = d • Sqrt (49 + 9) = d • Sqrt 58 = d

  7. Determine if the triangle formed by the coordinates (-2, 1), (2, 3), and (3,1) is an isosceles triangle. • Find the length of each side by using the distance formula. Sqrt((-2 – 2)2 + (1 – 3)2)) = Sqrt 20 Sqrt((2 – 3)2 + (3 – 1)2) = Sqrt 5 Sqrt((3 - -2)2 + (1 – 1)2 = Sqrt 25 = 5 Not isosceles since no two sides are equal. If you test pythagorean theorem you will find that it is a right triangle. (sqrt 20)2 + (sqrt 5)2 = (5)2

  8. Midpoint Formula • To find the midpoint of a line segment, average the x-coordinates and average the y-coordinates of the endpoints. • M(x, y) = ( (x1 + x2)/2 , (y1 + y2)/2)) • (-5, 5) to (3, 1) = (-5 + 3)/2 , (5 + 1)/2 • (-2/2, 6/2) => (-1, 3)

  9. Verify that the following is a right triangle, then find the area. • (4, -3), (0, -3), (4, 2) • Sqrt ((4 - 0)2 + (-3 - -3)2) = sqrt 16 = 4 • Sqrt ((0 – 4)2 + (2 - -3)2) = sqrt (41) • Sqrt ((4 – 4)2 + (2 - -3)2) = sqrt (25) = 5 • Right triangle if a2 + b2 = c2. When testing remember that ‘a’ and ‘b’ are the legs of the triangle, the shorter sides. • Does (sqrt 25)2 + (sqrt 16)2 = (sqrt 41)2? • Yes to it is a right triangle. • Area = ½ l w so Area = .5 (5) (4) = 10

  10. Assignment • Page 163 • #11, 13, 17, 21, 27, 33, 37, 43, 45, 53, 61, 65

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