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Distance and Midpoint Formulas in Algebra

Understand how to calculate distances and midpoints using algebraic formulas, including examples and exercises. Practice Pythagorean Theorem and midpoint calculations to enhance your math skills. Get ready to master distance and midpoint concepts in algebra!

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Distance and Midpoint Formulas in Algebra

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  1. ALGEBRA 12.6 The Distance and Midpoint Formulas

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

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

  4. Example Decide whether the points (6,4), (-3,1) and (9,-5) are vertices of a right triangle. √ √ = d1 = + (6 + 3)² (4 – 1)² 81 + 9 =√90 = 3√10 √ √ = d2 = + = √180 = 6√5 (-3 - 9)² (1 + 5)² 144 + 36 √ √ = d3 = + = √90 = 3√10 (6 - 9)² (4 + 5)² 9 + 81 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 short² long² (3√10)² + (3√10)² = (6√5)² 90 + 90 = 180 Yes. It is a right triangle. 180 = 180

  5. The Midpoint Formula The midpoint of the segment that joins points (x1,y1) and (x2,y2) is the point • (6,8) • (1,5) • (-4,2)

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

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

  8. Exercise There are 90 feet between consecutive bases on a baseball diamond. Suppose 3rd 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. 2nd ● 3rd 1st (10,0) (100,-90) 10 + 100 0 - 90 midpoint: , home 2 2 (55,-45)

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

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