1. Mrs. McConaughyGeometry1 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

2. Mrs. McConaughyGeometry2 PART I: FINDING DISTANCE

3. Mrs. McConaughyGeometry3 The Coordinate Plane Quadrant I (+, +) Quadrant II (-, +) Quadrant III (-, -)Quadrant IV (+, -) (0,0) T The coordinates of point T are ________. (6,3) Origin  The Coordinate Plane

4. Mrs. McConaughyGeometry4 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.

5. Mrs. McConaughyGeometry5 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 = ___) Unfortunately, this counting approach does NOT work for EF which is a diagonal segment. 7 3

6. Mrs. McConaughyGeometry6 Method 2: To find the distance between two points, A(x 1, y 1 ) and B(x 2, y 2 ), that are not on a horizontal or vertical line, we can use the Distance Formula. FormulaThe Distance Formula The distance, d, between two points, A(x 1, y 1 ) and B(x 2, y 2 ), is Alert! The Distance Formula can be used for all line segments: vertical, horizontal, and diagonal.

7. Mrs. McConaughyGeometry7 Finding Distance 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. (0,0) (6,8) (0,0)(6,8) ALERT! Order is important when using Distance Formula.

8. Mrs. McConaughyGeometry8 Example: Given (0,0) and (6,8), find the distance between the two points.

9. Mrs. McConaughyGeometry9 Applying the Distance Formula (2,4)Jackson (__,__) Symphony (__,__) City Plaza(__,__) Cedar (__,__) Central (__,__) North (__,__) Oak Each morning H. I. Achiever takes the “bus line” from Oak to Symphony. How far is the bus ride from Oak to Symphony?

10. Mrs. McConaughyGeometry10

11. Mrs. McConaughyGeometry11 Final Checks for Understanding 1.State the Distance Formula in words. 2.When should the Distance Formula be used when determining the distance between two given points? 3.Find the length of segment AB given A (-1,-2) and B (2,4).

12. Mrs. McConaughyGeometry12 Homework Assignment Page 46, text: 1-17 odd. *Extra Practice WS: Distance Formula with Solutions Available Online

13. Mrs. McConaughyGeometry13 PART II: FINDING THE MIDPOINT OF A SEGMENT

14. Mrs. McConaughyGeometry14 Vocabulary midpoint of a segment - _______________ __________________________________ __________________________________ point on a segment which divides the segment into two congruent segments

15. Mrs. McConaughyGeometry15 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.

16. Mrs. McConaughyGeometry16 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.

17. Mrs. McConaughyGeometry17 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 (x 1, y 1 ) and (x 2, y 2 ) has coordinates The Midpoint Formula works for all line segments: vertical, horizontal or diagonal.

18. Mrs. McConaughyGeometry18 Finding the Midpoint Find the midpoint of line segment AB. A (-3,4) B (2,1) Check your answer here: here

19. Mrs. McConaughyGeometry19 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.

20. Mrs. McConaughyGeometry20 Solve algebraically: M(-1,1), C(1,-3) and D(x,y) Substitute into the Midpoint Formula:

21. Mrs. McConaughyGeometry21 Solve for each variable separately: (-3,5)

22. Mrs. McConaughyGeometry22 Other Methods of Solution: 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.)

23. Mrs. McConaughyGeometry23 Final Checks for Understanding 1.Name two ways to find the midpoint of a given segment. 2.What method for finding the midpoint of a segment works for all lines…horizontal, vertical, and diagonal? 3.Explain how to find the coordinates of an endpoint when you are given an endpoint and the midpoint of a segment.

24. Mrs. McConaughyGeometry24 Homework Assignment: Page 46, text: 1-17 odd. *Extra Practice WS: Midpoint Formula with Solutions Available Online

25. Mrs. McConaughyGeometry25 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: Click here to return to lesson.here