1. Geometry Notes Section 1-3 9/7/07

2. What you’ll learn • How to find the distance between two points given the coordinates of the endpoints. • How to find the coordinate of the midpoint of a segment given the coordinates of the endpoints. • How to find the coordinates of an endpoint given the coordinates of the other endpoint and the midpoint.

3. Vocabulary Terms: • Midpoint • Segment bisector

4. M Q P Midpoint • In general the midpoint is the exact middle point in a line segment, but how do we define it geometrically? • If M is going to be the midpoint of PQ, then what rules does it have to follow?

5. M Q P Geometric definition of a segment’s midpoint. . . • Does the midpoint have to be located anywhere special? • YUP • Between the endpoints P and Q. • Rule #1: M must be between P and Q. • Remember this implies collinearity • And PM + MQ = PQ

6. M Q P Any other requirements for midpoint? • Yup— • It has to cut the segment in half. How do we express that geometrically? • In half means in two equal pieces. . . • Equal pieces—Equal length or CONGRUENT • Rule #2: • PM = MQ or PMMQ.

7. Can you identify and model a segment’s midpoint? • How do you model/illustrate equal length or congruence? • Identical markings on congruent parts/pieces.

8. Now to find the length of the segment or distance between the endpoints. . . . • First consider a simple number line. • Then we’ll look at the coordinate plane.

9. Finding the distance between 2 pts on a number line. • Use the coordinates of a line segment to find its length. • Consider a simple number line: P Q -3 -2 -1 0 1 2 3 4 5 6 • How would you find PQ?

10. To find the distance between two points on a number line: • Subtract the coordinates then take the absolute of that number (remember distance can’t be negative).

11. One dimensional – piece of cake. . What happens with 2-dimensions? 2-Dimensional refers to a coordinate plane

12. How to find distance on a coordinate plane • There are two methods • Pythagorean theorem • Distance Formula

13. Everyone knows the Pythagorean theorem. . . . • a2 + b2 = c2 • Where a, b, and c refer to the sides of a RIGHT triangle. . . • How do we get a right triangle out of a line segment?

14. AB= 5 • a2 + b2 = c2 • 42 + 32 = (AB)2 • 16 + 9 = (AB)2 • 25 = (AB)2 • 5 = AB a = 4 b = 3

15. In order to use the Pythagorean theorem. . . . • You have to complete the right triangle. What if the numbers are too big to graph? • There has to be another way. . .

16. The Distance Formula • The distance between two points with coordinates (x1, y1) and (x2, y2) • Using the same segment in our earlier example. . . .

17. The distance between two points with coordinates A(-2, -1) and B(1, 3) Look familiar???

18. There is a relationship between the Pythagorean Theorem and the Distance Formula. . . . • If you solve a2 + b2 = c2 for c, you will get • a and b represent the vertical and horizontal distances from the right triangle • vertical distance = subtracting the y-coordinates • horizontal distance = subtracting the x-coordinates

19. So. . . . • The distance formula related to the Pythagorean theorem because. . .

20. Can you find distance on a coordinate plane? • Using both methods? • Pythagorean theorem • Distance Formula a2 + b2 = c2

21. P Q -3 -2 -1 0 1 2 3 4 5 6 Finding the location (coordinate) of the midpoint • On a number line. . . . • Recall the midpoint is exactly half way between the endpoints of a segment • At what coordinate is the midpoint of PQ located? • The midpoint would be located at 2.5

22. P Q -3 -2 -1 0 1 2 3 4 5 6 Finding the location (coordinate) of the midpointmathematically • On a number line. . . . • The coordinate of the midpoint is the average of the coordinates of the endpoints • HUH?

23. Average the coordinates of the endpoints. . . . • Formula: • a is the coordinate of one endpoint • bis the coordinate of the other endpoint

24. P Q -3 -2 -1 0 1 2 3 4 5 6 Back to our example. . . . • Formula: • 1 is the coordinate of one endpoint • 4is the coordinate of the other endpoint

25. Finding the location (coordinate) of the midpointon a coordinate plane • Basically it’s the same as finding the midpoint on a number line • Recall the midpoint is exactly half way between the endpoints of a segment • We averaged the coordinates for a number line and we will average the coordinates for a coordinate plane

26. Average the coordinates of the endpoints. . . . • Formula: • (x1, y1) is the coordinate of one endpoint • (x2, y2) is the coordinate of the other endpoint

27. Find the coordinate of the midpoint of AB.

28. We know: A(-2, -1) B(1, 3) Formula: Fill It In: Simplify It:

29. Find the coordinate of the missing endpoint…

30. We know (xm, ym) is (1, 1) and (x1, y1)is (-2, -1) Formula: Fill It In: Split It:

31. Solve for x2:

32. Solve for y2:

33. FINALLY our answer is . . . . (4, 3)

34. Have you learned. . . • How to find the distance between two points given the coordinates of its endpoints? • How to find the coordinate(s) of the midpoint of a segment given the coordinates of the endpoints? • How to find the coordinates of an endpoint given the coordinates of the other endpoint and the midpoint? Assignment: Worksheet 1.3