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The Coordinate Plane

The Coordinate Plane. During this lesson you will: Find the distance between two points in the plane Find the coordinates of the midpoint of a segment. PART I: FINDING DISTANCE. The Coordinate Plane. Quadrant II (-, +). Quadrant I (+, +). T. The coordinates of point T are ________.

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The Coordinate Plane

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  1. The Coordinate Plane During this lesson you will: Find the distance between two points in the plane Find the coordinates of the midpoint of a segment Geometry

  2. PART I: FINDING DISTANCE Geometry

  3. The Coordinate Plane Quadrant II (-, +) Quadrant I (+, +) T The coordinates of point T are ________. (6,3) (0,0) Origin  Quadrant III (-, -) Quadrant IV (+, -) The Coordinate Plane Geometry

  4. When working with Coordinate Geometry, there are many ways to find distances (lengths) of line segments on graph paper. Let's examine some of the possibilities: Method 1: Whenever the segments are horizontal or vertical, the length can be obtained by counting. Geometry

  5. Unfortunately, this counting approach does NOT work for EF which is a diagonal segment. Method One Whenever the segments are horizontal or vertical, the length can be obtained by counting. When we need to find the length (distance) of a segment such as AB, we simply COUNT the distance from point A to point B.(AB= ___) We can use this same counting approach for CD .(CD= ___) 7 3 Geometry

  6. Method 2: To find the distance between two points, A(x1, y1) and B(x2, y2), that are not on a horizontal or vertical line, we can use theDistance Formula. Alert! The Distance Formula can be used for all line segments: vertical, horizontal, and diagonal. Geometry

  7. Finding Distance ALERT! Order is important when using Distance Formula. What is the distance between the two points on the right? STEP 1: Find the coordinates of the two points.____________ STEP 2: Substitute into the Distance Formula. (6,8) (0,0) (6,8) (0,0) Geometry

  8. Example:Given (0,0) and (6,8), find the distance between the two points. Geometry

  9. Applying the Distance Formula Each morning H. I. Achiever takes the “bus line” from Oak to Symphony. How far is the bus ride from Oak to Symphony? (2,4)Jackson (__,__) North (__,__) Central (__,__) Symphony (__,__) Cedar (__,__) City Plaza (__,__) Oak Geometry

  10. Geometry

  11. Final Checks for Understanding • State the Distance Formula in words. • When should the Distance Formula be used when determining the distance between two given points? • Find the length of segment AB given A (-1,-2) and B (2,4). Geometry

  12. Homework Assignment Page 46, text: 1-17 odd. *Extra Practice WS: Distance Formula with Solutions Available Online Geometry

  13. PART II: FINDING THE MIDPOINT OF A SEGMENT Geometry

  14. Vocabulary midpoint of a segment - ___________________________________________________________________________________ point on a segment which divides the segment into two congruent segments Geometry

  15. In Coordinate Geometry, there are several ways to determine the midpoint of a line segment. Method 1: If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.  Geometry

  16. Method 1: Horizontal or Vertical Lines If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints. Geometry

  17. The Midpoint Formula works for all line segments:  vertical, horizontal or diagonal. To find the coordinates of the midpoint of a segment when the lines are diagonal, we need to find the average (mean) of the coordinates of the midpoint. The Midpoint Formula: The midpoint of a segment endpoints (x1 , y1) and (x2 , y2) has coordinates Geometry

  18. Finding the Midpoint Find the midpoint of line segment AB. A (-3,4) B (2,1) Check your answer here: Geometry

  19. Consider this “tricky” midpoint problem: M is the midpoint of segment CD.  The coordinates M(-1,1) and C(1,-3) are given.  Find the coordinates of point D. First, visualize the situation.  This will give you an idea of approximately where point D will be located.  When you find your answer, be sure it matches with your visualization of where the point should be located. Geometry

  20. Solve algebraically:M(-1,1), C(1,-3) and D(x,y)Substitute into theMidpoint Formula: Geometry

  21. Solve for each variable separately: (-3,5) Geometry

  22. Verbalizing the algebraic solution: Some students like to do these "tricky" problems by just examining the coordinates and asking themselves the following questions:"My midpoint's x-coordinate is -1.  What is -1 half of? (Answer -2)What do I add to my endpoint's x-coordinate of +1 to get -2? (Answer -3)This answer must be the x-coordinate of the other endpoint."These students are simply verbalizing the algebraic solution.  (They use the same process for the y-coordinate.) Geometry

  23. Final Checks for Understanding • Name two ways to find the midpoint of a given segment. • What method for finding the midpoint of a segment works for all lines…horizontal, vertical, and diagonal? • Explain how to find the coordinates of an endpoint when you are given an endpoint and the midpoint of a segment. Geometry

  24. Homework Assignment: Page 46, text: 1-17 odd. *Extra Practice WS: Midpoint Formula with Solutions Available Online Geometry

  25. Solution Given: A(-3,4); B(2,1) The midpoint will have coordinates: Alert! Your answer may contain a fraction.  Answers may be written in fractional or decimal form. Answer: Geometry Click here to return to lesson.

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