Question

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 In the standard (x,y) coordinate plane A,B and C are collinear. What is y-coordinate of point C?

 

 

Answer 1

Let the points be A(x₁,y₁), B(x₂,y₂), C(x₃,y₃). We need y₃.

The slope of AB is (y₂-y₁)/(x₂-x₁), which is the same slope as AC and BC. We don't know how far away C is from A or B, but we do know that C lies on the line AB or AB extended.

So y-y₁=[(y₂-y₁)/(x₂-x₁)](x-x₁) is the equation of AB, that is, y=(y₂-y₁)(x-x₁)/(x₂-x₁)+y₁.

This can be written y=[(y₂-y₁)(x-x₁)+y₁(x₂-x₁)]/(x-x₁),

y=(xy₂-x₁y₂-xy₁+x₂y₁)/(x-x₁).

C can be any point on this line. If x=a, we have the point C(a,y₃) where:

y₃=(ay₂-x₁y₂-ay₁+x₂y₁)/(a-x₁),

y₃=(a(y₂-y₁)-x₁y₂+x₂y₁)/(a-x₁).

By plugging in the value of a and the coordinates of A and B, y₃ can be calculated.