Determine if (8, 3), (5, 2), and (2, 1) are collinear

[SCSmith]

Determine if points are Colinear.

(8,3), (5,2), (2,1)

sqrt (5 - 8)^2 + (2 - 3)^2
sqrt (-3)^2 + (-1)^2
sqrt 9 + 1
sqrt 10

sqrt (2 - 5)^2 + (1 - 2)^2
sqrt (-3)^2 + (-1)^2
sqrt 9 + 1
sqrt 10

sqrt (2 - 8)^2 + (1 - 3)^2
sqrt (-6)^2 + (-2)^2
sqrt 36 + 4
sqrt 40
sqrt 10 + sqrt 10 = 2(sqrt 10)
sqrt 40 = 2(sqrt 10)

so they are collinear.

Is this logical?

 

[soroban]

Re: Distance Formula

Hello, SCSmith

Determine if points are collinear: \(\displaystyle A(8,3),\;B (5,2),\;C (2,1)\)

Your work and reasoning are absolutely correct.
. . But this can be done with slopes, too.

Slope of \(\displaystyle AB\) is: \(\displaystyle \:m_{_{AB}}\:=\:2\,-\,3}{5\,-\,8}\:=\:\frac{-1}{-3}\:=\:\frac{1}{3}\)

Slope of \(\displaystyle BC\) is: \(\displaystyle \:m_{_{BC}} \:=\:\frac{1\,-\,2}{2\,-\,5}\:=\:\frac{-1}{-3}\:=\:\frac{1}{3}\)

Therefore: A, B, and C are collinear.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Another method (a bit longer, though).

Find the equation of the line through \(\displaystyle A\) and \(\displaystyle B\).

Then show that point \(\displaystyle C\) lies on that line.
. . That is, the coordinates of \(\displaystyle C\) satisfy the equation.