ALGEBRA 12 6 The Distance and Midpoint Formulas
ALGEBRA 12. 6 The Distance and Midpoint Formulas
The Distance Formula The distance d between points and is: Why? Let’s try an example to find out! Find the distance between (– 3, 4) and (1, – 4). (-3, 4) . 4 8 4√ 5 Pythagorean Theorem! . (1, -4)
Examples Find the distance between the two points. Leave answers in simplified radical form. 1. A (3, 5) B (7, 8) d = √ (7 – 3)² + Distance of AB = 5 (8 – 5)² 2. C (-7, 2) D (-2, -10) = √ 16 + 9 =√ 25 = 5 Distance of CD = 13 d = √ (-2 +7)² + (-10 – 2)² = √ 25 + 144 =√ 169 = 13
Example Decide whether the points (6, 4), (-3, 1) and (9, -5) are vertices of a right triangle. d 1 = √ (6 + 3)² + d 2 = √ (-3 - 9)²+ (4 – 1)² (1 + 5)² = √ 81 + 9 =√ 90 = 3√ 10 = √ 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? short² long² (3√ 10)² + (3√ 10)² = (6√ 5)² 90 + 90 = 180 Yes. It is a right triangle. 180 = 180
The Midpoint Formula The midpoint of the segment that joins points (x 1, y 1) and (x 2, y 2) is the point • (1, 5) • (-4, 2) • (6, 8)
How does it work? Find the coordinate of the Midpoint of BC. B (12, 7) 7 A ● 4 1 B ● ● C● 4 8 12 C (4, 1) Midpoint: 12 + 4 7+1 2 2 , (8, 4)
Exercises 1. A (3, 5) B (7, -5) midpoint: 3+7 5+(-5) 2 , 2 (5, 0) 2. A (0, 4) B (4, 3) midpoint: 0+4 4+3 2 , 2 (2, 7 2 )
Exercise 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? midpoint: Sketch it. 2 nd 3 rd ● 10 + 100 0 - 90 2 , 2 (55, -45) (10, 0) home 1 st (100, -90)
Homework pg. 748 #15 -45 odd #54, 55
- Slides: 9