1. Obj. 7 Midpoint and Distance Objectives: The student is able to (I can): Find the midpoint of two given points. Find the coordinates of an endpoint given one endpoint and a midpoint. Find the distance between two points.

2. The coordinates of a midpoint are the averages of the coordinates of the endpoints of the segment. C T 3. The coordinates of a midpoint are the averages of the coordinates of the endpoints of the segment. 1 3 2 1 2 2 + = = C A T 4. -2 2 4 6 8 10 -2 2 4 6 8 10 x y D G 5. -2 2 4 6 8 10 -2 2 4 6 8 10 x y x-coordinate: 2 8 10 5 2 2 + = = D G 6. -2 2 4 6 8 10 -2 2 4 6 8 10 x y x-coordinate: y-coordinate: 2 8 10 5 2 2 + = = 4 8 12 6 2 2 + = = D G 7. -2 2 4 6 8 10 -2 2 4 6 8 10 x y x-coordinate: y-coordinate: 2 8 10 5 2 2 + = = 4 8 12 6 2 2 + = = (5, 6) D O G 8. midpoint formula The midpoint M of with endpoints A(x1, y1) and B(x2, y2) is found by AB 0 A B x1 x2 y1 y2 9. midpoint formula The midpoint M of with endpoints A(x1, y1) and B(x2, y2) is found by AB 1 12 2 M , 2 2 yxx y+ + 0 A B x1 x2 y1 y2 M average of x1 and x2 average of y1 and y2 10. Example Find the midpoint of QR for Q(3, 6) and R(7, 4) 11. Example Find the midpoint of QR for Q(3, 6) and R(7, 4) x1 y1 x2 y2 Q(3, 6) R(7, 4) 12. Example Find the midpoint of QR for Q(3, 6) and R(7, 4) x1 y1 x2 y2 Q(3, 6) R(7, 4) 21x 3x 7 4 2 2 2 2 + + = = = 13. Example Find the midpoint of QR for Q(3, 6) and R(7, 4) x1 y1 x2 y2 Q(3, 6) R(7, 4) 21x 3x 7 4 2 2 2 2 + + = = = 21 2 1 y 2 2 y 6 2 4+ + = = = 14. Example Find the midpoint of QR for Q(3, 6) and R(7, 4) x1 y1 x2 y2 Q(3, 6) R(7, 4) 21x 3x 7 4 2 2 2 2 + + = = = 21 2 1 y 2 2 y 6 2 4+ + = = = M(2, 1) 15. Problems 1. What is the midpoint of the segment joining (8, 3) and (2, 7)? A. (10, 10) B. (5, 2) C. (5, 5) D. (4, 1.5) 16. Problems 1. What is the midpoint of the segment joining (8, 3) and (2, 7)? A. (10, 10) B. (5, 2) C. (5, 5) D. (4, 1.5) + = = 8 2 10 5 2 2 + = = 3 7 10 5 2 2 17. Problems 2. What is the midpoint of the segment joining (4, 2) and (6, 8)? A. (5, 5) B. (1, 3) C. (2, 6) D. (1, 3) 18. Problems 2. What is the midpoint of the segment joining (4, 2) and (6, 8)? A. (5, 5) B. (1, 3) C. (2, 6) D. (1, 3) + = = 4 6 2 1 2 2 ( )+ = = 2 8 6 3 2 2 19. If we are given an endpoint and the midpoint, we can use the distances between them to locate the missing endpoint. Example: T is the midpoint of SP. S has coordinates (5, 7) and T is at (2, 5). Find the coordinates of P. 20. If we are given an endpoint and the midpoint, we can use the distances between them to locate the missing endpoint. Example: T is the midpoint of SP. S has coordinates (5, 7) and T is at (2, 5). Find the coordinates of P. midpoint endpoint = distance x-coordinates: 2 5 = 3 21. If we are given an endpoint and the midpoint, we can use the distances between them to locate the missing endpoint. Example: T is the midpoint of SP. S has coordinates (5, 7) and T is at (2, 5). Find the coordinates of P. midpoint endpoint = distance x-coordinates: 2 5 = 3 y-coordinates: 5 (7) = 12 22. If we are given an endpoint and the midpoint, we can use the distances between them to locate the missing endpoint. Example: T is the midpoint of SP. S has coordinates (5, 7) and T is at (2, 5). Find the coordinates of P. midpoint endpoint = distance x-coordinates: 2 5 = 3 y-coordinates: 5 (7) = 12 new endpoint = midpoint + distance P(2 + 3, 5 + 12) 23. If we are given an endpoint and the midpoint, we can use the distances between them to locate the missing endpoint. Example: T is the midpoint of SP. S has coordinates (5, 7) and T is at (2, 5). Find the coordinates of P. midpoint endpoint = distance x-coordinates: 2 5 = 3 y-coordinates: 5 (7) = 12 new endpoint = midpoint + distance P(2 + 3, 5 + 12) = P(1, 17) 24. Problem 3. Point M(7, 1) is the midpoint of , where A is at (14, 4). Find the coordinates of point B. A. (7, 2) B. (14, 4) C. (0, 6) D. (10.5, 1.5) AB 25. Problem 3. Point M(7, 1) is the midpoint of , where A is at (14, 4). Find the coordinates of point B. A. (7, 2) B. (14, 4) C. (0, 6) D. (10.5, 1.5) AB = 7 14 7 = 1 4 5 ( ) ( )+ + = B 7 ( 7), 1 ( 5) B 0, 6 26. Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. + = = +2 2222 2 or c ( acb ba ) y x a b c = +2 22 c a b = + 22 c a b 22 164 93= + = + 25 5= = 27. Length of a = Length of b = y x a b c 2 1x x 2 1y y 28. Length of a = Length of b = so y x a b c 2 1x x 2 1y y ( ) ( ) 2 22 2 1 2 1c x x y y= + 29. Length of a = Length of b = so or y x a b c 2 1x x 2 1y y ( ) ( ) 2 22 2 1 2 1c x x y y= + ( ) ( )= + 2 2 2 1 2 1c x x y y 30. distance formula Given two points (x1, y1) and (x2, y2), the distance between them is given by Example: Use the Distance Formula to find the distance between F(3, 2) and G(-3, -1) ( ) ( ) 2 1 2 2 2 1d xx y y= + 31. distance formula Given two points (x1, y1) and (x2, y2), the distance between them is given by Example: Use the Distance Formula to find the distance between F(3, 2) and G(-3, -1) ( ) ( ) 2 1 2 2 2 1d xx y y= + x1 y1 x2 y2 3 2 3 1 ( ) ( )= + 2 2 F 33 1G 2 32. distance formula Given two points (x1, y1) and (x2, y2), the distance between them is given by Example: Use the Distance Formula to find the distance between F(3, 2) and G(-3, -1) ( ) ( ) 2 1 2 2 2 1d xx y y= + x1 y1 x2 y2 3 2 3 1 ( ) ( )= + 2 2 F 33 1G 2 ( ) ( )2 2 6 3 36 9= + = + = = 45 3 5 6.7 Note: Remember that the square of a negative number is positivepositivepositivepositive. 33. Problems 1. Find the distance between (9, 1) and (6, 3). A. 5 B. 25 C. 7 D. 13 34. Problems 1. Find the distance between (9, 1) and (6, 3). A. 5 B. 25 C. 7 D. 13 ( ) ( )= + 22 d 6 9 3 ( 1) 35. Problems 1. Find the distance between (9, 1) and (6, 3). A. 5 B. 25 C. 7 D. 13 ( ) ( ) ( ) ( ) = + = + = 22 2 2 d 6 9 3 ( 1) 3 4 5 36. Problems 2. Point R is at (10, 15) and point S is at (6, 20). What is the distance RS? A. 1 B. C. 41 D. 6.5 41 37. Problems 2. Point R is at (10, 15) and point S is at (6, 20). What is the distance RS? A. 1 B. C. 41 D. 6.5 41 ( ) ( )= + 2 2 d 6 10 20 15 38. Problems 2. Point R is at (10, 15) and point S is at (6, 20). What is the distance RS? A. 1 B. C. 41 D. 6.5 41 ( ) ( ) ( ) = + = + = + = 2 2 2 2 d 6 10 20 15 4 5 16 25 41