How to prove dot product without vectors?

Question

The dot product to find the distance from a point ($x_0, y_0$) to line $ax + by + c = 0$ is $d=\frac{|a(x_0)+b(y_0)+c|}{\sqrt{a^2+b^2}}$. I need to prove this formula but I am not supposed to use vectors. How else could I prove this?

Answer 1

Hint:

Let $A(x_1,y_1) $ and $B (x_2,y_2) $ lies on the line $ax+by+c=0$ and calculate the area of triangle $ABT$ where $T=(x_0,y_0) $ on two ways.

Further, If $ab\ne 0$ then you can take $x_1=0$ so $y_1 = -c/b $ and $y_2=0$ so $x_2=-c/a $. So you get $$AB = {c\over ab}\sqrt{a^2+b^2} $$

Use that $$ S= |D| /2$$ where $D $ is a determinant of the triangle $ABT $ and $S= d\cdot AB/2$.