How would you work out the perpendicular distance from the point 4 -2 to the straight line of 2x -y -5 equals 0?

Answer 1

Points: (4, -2)

Equation: 2x -y -5 = 0

Perpendicular equation: x+2y = 0

Both equations intersect at: (2 -1)

Prependicular distance is the square root of (4-2)2+(-2--1)2

So the distance is the square root of 5

Knowing the equation of the line, you can work out the gradient of a line perpendicular to the line:

Given the line in the form y = mx + c (where m is the gradient of the line and c is the y-intercept), the gradient of a perpendicular line (m') is given by:

mm' = -1 → m' = -1/m

The equation of the line perpendicular to the given line through a given point (xo, yo) can then be found by:

y - yo = m'(x - xo) = -1/m(x - xo)

With the two lines you can find their point of intersection, namely the point (xi, yi) that simultaneously satisfies both equations, and then use Pythagoras to find the distance from this point to the given point:

distance = √((xo - xi)2 + (yo - yi)2)