Collinear Points

Geometry Level 1

If three points $A=(k,k+2),$ $B=(0,k-6)$ and $C=(k-4,k)$ all lie on the same line, what is the value of $k?$

-4 -16 16 4

2 solutions

Chew-Seong Cheong
Sep 28, 2015

If $A$ , $B$ and $C$ are on a same straight line, then gradient of any two of the points is the same. Therefore,

The gradient of $AB$ is $\dfrac{k+2-(k-6)}{k-0} = \dfrac{8}{k}$
The gradient of $AC$ is $\dfrac{k+2-k}{k-(k-4)} = \dfrac{2}{4} = \dfrac{1}{2}$

$\Rightarrow \dfrac{8}{k} = \dfrac{1}{2} \quad \Rightarrow k = \boxed{16}$

if k=16 then A,B, and C isn't lying on the same line because when you substitute k neither of all the x or y values are equal. When you say they're on the same line either they have the same y-value or x-value so k not=16

Caeo Tan - 5 years, 8 months ago

No, if they have either the same $x$ value or $y$ value, they are perpendicular to each other. Check out the graph in my solution.

Chew-Seong Cheong - 5 years, 8 months ago

Joe Potillor
Feb 8, 2017

Collinear Points

2 solutions

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