1. ALGEBRA 12.6 The Distance and Midpoint Formulas

2. The distance d between points and is: Find the distance between (–3, 4) and (1, –4). Why? Let’s try an example to find out! (-3, 4).. (1, -4) 4 8 Pythagorean Theorem! 4√5 The Distance Formula

3. Examples 1. A (3,5) B (7,8) d = Distance of AB = 5 2. C (-7,2) D (-2,-10) Find the distance between the two points. Leave answers in simplified radical form. √ (7 – 3)² + (8 – 5)² √ 16 + 9 = =√25 = 5 d = √ (-2 +7)² + (-10 – 2)² √ 25 + 144 = =√169 = 13 Distance of CD = 13

4. Example d 1 = Yes. It is a right triangle. Decide whether the points (6,4), (-3,1) and (9,-5) are vertices of a right triangle. √ (6 + 3)² + (4 – 1)² √ 81 + 9 = =√90 = 3√10 d 2 = √ (-3 - 9)² + (1 + 5)² √ 144 + 36 = = √180 = 6√5 d 3 = √ (6 - 9)² + (4 + 5)² √ 9 + 81 = = √90 = 3√10 Now use the Pythagorean Theorem Converse to check. Does the sum of the squares of the two shorter sides equal the square of the longest side? (3√10)² + (3√10)² = (6√5)² 90 + 90 = 180 short² long² 180 = 180

5. The Midpoint Formula The midpoint of the segment that joins points (x 1,y 1 ) and (x 2,y 2 ) is the point (-4,2) (6,8) (1,5)

6. How does it work? 48 Find the coordinate of the Midpoint of BC. 12 7 ● 1 4 C ● AB 12 + 4 2, 7 + 1 2 (8,4) ● ● B (12,7) C (4,1) Midpoint:

7. Exercises 1. A (3,5) B (7,-5) midpoint: 3+7 2, 5+(-5) 2 (5,0) 2. A (0,4) B (4,3) midpoint: 0+4 2, 4+3 2 (2, ) 7 2

8. (55,-45) Exercise midpoint: 10 + 100 2, There are 90 feet between consecutive bases on a baseball diamond. Suppose 3 rd base is located at (10,0) and first base is located at (100,-90). A ball is hit and lands halfway between first base and third base. Where does the ball land? Sketch it. home 1st 2nd 3rd (10,0) (100,-90) ● 0 - 90 2

9. Homework pg. 748 #15-45 odd #54,55