Length of Perpendicular from a Point on a Line

Length of the perpendicular lines

A perpendicular is a line that intersects another line at a right angle of 90 degrees. To illustrate 90 degrees (also known as “right angle”) between two vertical lines, you may see it represented by a little square. Right-angle intersection means that the two lines are perpendicular.

The two main properties of the vertical line are: 

• Vertical lines always intersect or intersect. 
• The angle between any two vertical lines is always equal to 90 °

The triangle formula, the distance formula, and the area formula are all used to develop a formula for calculating the distance between two points on a line. Following the rules of Euclidean geometry, the distance between a point and a line may be conceived of as the shortest distance between a given point and any other point along an infinite straight line. The length of the line segment from the point to the nearest point on the line is the shortest distance between the point and the nearest point on the line, which is the perpendicular distance between the point and line.

Formulas

Use the area of ​​the distance equation and the triangle equation to derive an equation that measures the distance of a point from a line. 

Consider the line L in the XY plane where K (x1, y1) is any point at a distance d from line L. This line is represented by Ax + By + C = 0. The distance `d’ from the line to the point is the length of the perpendicular drawn from K to L. The x and y sections can be represented as (-C / A) and (-C / B), respectively.

Line L touches the x-axis and y-axis at points B and A, respectively. KJ is the vertical distance from point K tangent to the base AB of ΔKAB at point J. Given the three points K, B, and A, the coordinates are given as follows:

K (x1, y1), B (x2, y2) and A (x3, y3)

(x2, y2) = ((-C / A), O) and (x3, y3) = (O, (-C / B))

Area of triangle = ½ * base * perpendicular height

Area (Δ KAB) = ½ AB × KJ

⇒ KJ = 2 × area (Δ KAB) / AB 

area A = ½ |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|

= ½ | x1 (0 – (-C/B)) + (−C/A) ((−C/B) − y1) +0 (y1 − 0)|

= ½ |(C/B) × x1 – C/A ((−C/B) -y1) + 0|

= ½ |(C/B) × x1 – C/A ((−C-By1)/B)|

= ½ |(C/B) × x1 + C2/AB + ((BCy1)/AB)|

= ½ |(C/B) × x1 + (C/A) × y1 + (C2/AB)|

= ½ |C( x1/B + y1/A + C/AB)|

By multiplying and dividing by AB 

= ½ |C(ABx1/AB2 + (ABy1)/BA2 + (ABC2)/(AB)2|

= ½ |CAx1/AB + CBy1/AB + C2/AB|

=½ |C/ (AB)|.|Ax1 + By1 + C| 

Now the distance from AB with the given coordinates can be calculated by

AB = ((x2 – x1)2 + (y2– y1)2)½

Here, A(x1,y1) = A(0, -C/B) and B(x2,y2) = B(-C/A,0)

AB = (((-C/A)2 – 0) + (0 – (-C/B)2))½

= ((C/A)2 + (C/B)2)½

Distance, AB = |C/AB| (A2 + B2)½

The distance of the perpendicular KJ = d = |Ax1 + By1 + C| / (A2 + B2)½

Conclusion

With these three formulas, you may create a formula for determining the distance between two points. By definition, in Euclidean geometry, it is the distance between two points that determines the shortest distance between them. The perpendicular distance from a given point to the line can be calculated by measuring the length of the line segment from that place to the nearest point on the line.