Label the points A,B,C
- A = (1,2)
- B = (4,5)
- C = (8,9)
Let's find the distance from A to B, aka find the length of segment AB.
We use the distance formula.
[tex]A = (x_1,y_1) = (1,2) \text{ and } B = (x_2, y_2) = (4,5)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(1-4)^2 + (2-5)^2}\\\\d = \sqrt{(-3)^2 + (-3)^2}\\\\d = \sqrt{9 + 9}\\\\d = \sqrt{18}\\\\d = \sqrt{9*2}\\\\d = \sqrt{9}*\sqrt{2}\\\\d = 3\sqrt{2}\\\\[/tex]
Segment AB is exactly [tex]3\sqrt{2}[/tex] units long.
Now let's find the distance from B to C
[tex]B = (x_1,y_1) = (4,5) \text{ and } C = (x_2, y_2) = (8,9)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(4-8)^2 + (5-9)^2}\\\\d = \sqrt{(-4)^2 + (-4)^2}\\\\d = \sqrt{16 + 16}\\\\d = \sqrt{32}\\\\d = \sqrt{16*2}\\\\d = \sqrt{16}*\sqrt{2}\\\\d = 4\sqrt{2}\\\\[/tex]
Segment BC is exactly [tex]4\sqrt{2}[/tex] units long.
Adding these segments gives
[tex]AB+BC = 3\sqrt{2}+4\sqrt{2} = 7\sqrt{2}[/tex]
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Now if A,B,C are collinear then AB+BC should get the length of AC.
AB+BC = AC
Let's calculate the distance from A to C
[tex]A = (x_1,y_1) = (1,2) \text{ and } C = (x_2, y_2) = (8,9)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(1-8)^2 + (2-9)^2}\\\\d = \sqrt{(-7)^2 + (-7)^2}\\\\d = \sqrt{49 + 49}\\\\d = \sqrt{98}\\\\d = \sqrt{49*2}\\\\d = \sqrt{49}*\sqrt{2}\\\\d = 7\sqrt{2}\\\\[/tex]
AC is exactly [tex]7\sqrt{2}[/tex] units long.
Therefore, we've shown that AB+BC = AC is a true equation.
This proves that A,B,C are collinear.
For more information, check out the segment addition postulate.
