What is the best solution to solve equation regarding straight line?

Question

given 3 points ,A=(1,1) B=(3,4) C=(7,10).show ABC lie on straight line.

i know solutionl like finding the slope but is it the most suitable way to find the answer?because 2 parallel lines also have same gradient..so i dont think its the best way to solve the question?any answer that is undeniable please tell me?another thing pls gimme answer that is still under topic function and grapth no cordinate geometry or vector.

Answer 1

Two parallel lines cannot intersect; so, here you cannot tell that AB and BC are parallel [as B is common point]

If you feel difficulty in using slope, use distance formula and easily you get that AC=AB+BC

Answer 2

I would find the equation for a straight line passing thru points A and B, and then see if point C lies on it.

y = mx + b

= (4-1)/(3-1)x + b

y = 3x/2 + b

find b:

1 = 3(1)/2 + b

b = -1/2

y = 3x/2 - 1/2

Try (7,10)

y = 3(7)/2 - 1/2

= 10

so C lies on the line through A and B

Answer 3

To me easiet way is using slope.

The slope between A,B should equal slope between A,C.

Slope of A,B = (4-1)/(3-1) = 3/2.

Slope of A,C = (10-1)/(7-1) = 9/6 = 3/2.

SO, they are on straight line since slopes are equal.