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Distance Formula

d = √ ( x 1 – x 2 ) 2 + ( y 1 – y 2 ) 2 , where d is the distance between the points ( x 1 , y 1 ) and ( x 2 , y 2 ). Distance Formula. d = √ ( x 1 – x 2 ) 2 + ( y 1 – y 2 ) 2. = √ (– 3 – 2) 2 + [5 – (– 6)] 2. = √ (– 5) 2 + 11 2. = √ 146. Example 1.

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Distance Formula

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  1. d = √(x1 – x2)2 + (y1 – y2)2, where d is the distance between the points (x1, y1) and (x2, y2). Distance Formula

  2. d = √(x1 – x2)2 + (y1 – y2)2 = √(–3 – 2)2 + [5 – (–6)]2 = √(–5)2 + 112 = √146 Example 1 Find the distance between the points (–3, 5) and (2, –6). ≈ 12.1 units

  3. A (x1, y1) C B (x1, y2) (x2, y2)

  4. The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is , . x1 + x2 2 y1 + y2 2 Midpoint Formula

  5. –8 + (–1)2 –2 + 52 , –92 32 = , Example 2 Find the midpoint of the segment extending from (–8, –2) to (–1, 5). = (–4.5, 1.5)

  6. Example 3 Parallelogram ABCD has vertices at A (3, 0), B (7, 1), C (7, 4), and D (3, 3). Find the length of each side and the midpoint of each diagonal.

  7. CD = √(7 – 3)2 + (4 – 3)2 AB = √(7 – 3)2 + (1 – 0)2 = √17 = √17 = √16 + 1 = √16 + 1 AD = y1 – y2 = 3 – 0 = 3 units BC = y1 – y2 = 4 – 1 = 3 units ≈ 4.1 units ≈ 4.1 units

  8. 102 102 42 42 0 + 42 1 + 32 7 + 32 3 + 72 , , , , = = = = Midpoint of AC = (5, 2) Midpoint of BD = (5, 2)

  9. √13 Example Find the distance between the points (–1, 6) and (2, 4).

  10. Example Find the midpoint of the segment extending from(4, 7) to (10, 13). (7, 10)

  11. Example If (–4, 5) is one endpoint of a line segment with midpoint (3, –2), what is the other endpoint? (10, –9)

  12. Example Graph an isosceles triangle with vertices at (0, 0), (8, 0), and (4, 6). Find the midpoint of each side of the triangle. (4, 0), (2, 3), (6, 3)

  13. Example Graph an isosceles triangle with vertices at (0, 0), (8, 0), and (4, 6). Is the figure formed by joining the midpoints similar to the original triangle? yes

  14. Exercise Use the distance formula to find the lengths of the diagonals AC and BD of rectangle ABCD, whose vertices are A (–1, 4), B (7, –2), C (4, –6), and D (–4, 0). How do these two lengths compare?

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