Question

5. Show that the following points are collinear. a) (1, 2), (4, 5), (8,9) ​

Answer 1

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.

`A = (x_1,y_1) = (1,2) \text{ and } B = (x_2, y_2) = (4,5)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((1-4)^2 + (2-5)^2)\\\\d = √((-3)^2 + (-3)^2)\\\\d = √(9 + 9)\\\\d = √(18)\\\\d = √(9*2)\\\\d = √(9)*√(2)\\\\d = 3√(2)\\\\`

Segment AB is exactly
`3√(2)` units long.

Now let's find the distance from B to C

`B = (x_1,y_1) = (4,5) \text{ and } C = (x_2, y_2) = (8,9)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((4-8)^2 + (5-9)^2)\\\\d = √((-4)^2 + (-4)^2)\\\\d = √(16 + 16)\\\\d = √(32)\\\\d = √(16*2)\\\\d = √(16)*√(2)\\\\d = 4√(2)\\\\`

Segment BC is exactly
`4√(2)` units long.

Adding these segments gives

`AB+BC = 3√(2)+4√(2) = 7√(2)`

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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

`A = (x_1,y_1) = (1,2) \text{ and } C = (x_2, y_2) = (8,9)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((1-8)^2 + (2-9)^2)\\\\d = √((-7)^2 + (-7)^2)\\\\d = √(49 + 49)\\\\d = √(98)\\\\d = √(49*2)\\\\d = √(49)*√(2)\\\\d = 7√(2)\\\\`

AC is exactly
`7√(2)` 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.