Distance of a Point From a Line

The distance of a point from a line is the smallest possible distance between the point and the line. Generally, It is feasible to draw an endless number of lines from a specific point to a line inside a plane by using the coordinate system. Here the question arises: how to point out the shortest distance between the point and the line, or how can we measure the distance between this line and the point.

What is the Distance of a Point From a Line?

It is the shortest distance between the line and the point. In order to measure the distance, just draw a perpendicular to the given line such that it passes through the given point and connects the line and the point.

When we talk about a triangle, especially a right-angled triangle, the hypotenuse of the triangle is the longest line segment in the triangle. In order to determine the height of the triangle, creating a line as the triangle’s altitude will help.  A right-angled triangle is formed when we draw the foot of the perpendicular from a given point to a line, as well as any other segment that connects the provided point to the line. Moreover, the hypotenuse of a triangle will always be the second line segment in the triangle. Now by using the distance of a point from a line formula, the distance between the point and the line can be calculated.

Perpendicular Distance of a Point From a Line

Learn to derive the distance of a point from a line formula mathematically. Following on from the previous paragraph, the shortest distance between a point and a line is the length of the perpendicular drawn between the provided point and the given line (or vice versa). Here are the steps to derive the formula to find out the shortest distance between the line and the point.

Step I- Consider the line L: Xx + Yy + Z= 0  whose distance from the point P (a1, b1) is d.

Step II- Sketch a perpendicular PN from the point P to line L.

Step III- Let S and T be the points from where the line meets the x,y axes.

Step IV- The location of the points are represented by the notation S(-C/X,0) and T(0,-C/Y)

Derivation of the Formula to Find Distance Between a Point and a Line

Solved Problems

Question 1:

Calculate the distance between point (-3, 5) and the line 4x – 3y – 26 = 0.

Solution->

The following is the equation of a line: : 4x – 3y – 26 = 0

Point: (-3, 5)

When they are compared to the usual forms, it becomes clear that

X = 4, Y = -3, C = -26

a1 = -3, b1 = 5

We are aware that the perpendicular distance (d) of a line Xa + Yb + C = 0 from a point (a1, b1) is given by

d = |Xa1 + Yb1 + C|]/ √(X2 + Y2)

Substituting the values results in

d = |4(-3) + (-3)(5) + (-26)|/√[(4)2 + (-3)2]

= |-12 – 15 – 26|/√(16 + 9)

= |-53|/√25

= 53/5

Hence, the required distance is 53/5 units.

Question 2:

Calculate the distance between point (5,1) and the line y = 3x + 1.

Solution: Given point (5,1) and line y = 3x + 1.

We need to find the distance between them.

y = 3x + 1

3x + 1 – y = 0

When they are compared to the usual forms, it becomes clear that

X = 3, Y = -1, C = 1

a1 = 5, b1 = 1

We know that the perpendicular distance (d) of a line Xa + Yb + C = 0 from a point (a1, b1) is given by

d = |Xa1 + Yb1 + C|]/ √(X2 + Y2)

Substituting the values results in

d = [|3.5 + (-1)(1) + 1|]/√(32 + (-1)2)

d = 15/√10

Conclusion

​​We use the distance formula and the area of the triangle formula to get the formula for measuring the distance of a point from a line. According to Euclidean geometry, the distance from a point to a line is the smallest distance between any two points on an infinite straight line. The length of the line segment from the point to the closest point on the line is the shortest distance from that point, which is the point’s perpendicular distance to the line.