Distance from Point to Line

According to Euclidean geometry, the distance from a point to a line is the smallest distance between two points on an infinite straight line. The length of the line segment from the point to the nearest point on the line is the smallest distance from that point, which is perpendicular to the line. The formula for estimating the distance between two points can be obtained and stated in a variety of ways. Knowing the distance from a point to a line may be beneficial in a variety of real-world scenarios, such as determining the distance between two objects, such as two trees.

Let us understand the meaning coordinate geometry,

Co-ordinate Geometry:

Coordinate geometry is a branch of mathematics that helps in displaying geometric structures on a two-dimensional plane and learning about their properties. To get a grasp of Coordinate geometry, we will first learn about the coordinate plane and the coordinates of a point.

The distance from a point to a line is the shortest distance between two points on an infinite straight line, according to Euclidean geometry. The length of the line segment from the point to the nearest point on the line is the smallest distance from that point, which is perpendicular to the line. The formula for estimating the distance between two points can be obtained and stated in a variety of ways. Knowing the distance from a point to a line may be beneficial in a variety of real-world scenarios, such as determining the distance between two objects, such as two trees.

Distance from point to any line

The shortest distance between two points is the distance between two lines. It is the shortest distance between two points on a straight line. This minimal length distance can be represented by a line segment perpendicular to the line.

• Point 

A point is a precise position in mathematics. The term “point” refers to a location rather than an item. A point can be represented on a coordinate axis, and its coordinates are defined by a pair of integers that specify its exact position.

• Line

Lines are an idealisation of such things, which are frequently characterised in terms of two points (for example, AB) or referred to with a single letter (e.g., l).

Coordinate geometry formulae make it easier to prove the many qualities of lines and figures represented by coordinate axes. Coordinate geometry formulae include the distance formula, slope formula, midpoint formula, section formula, and line equation.

The coordinate geometry formulae make it easier to prove the various properties. The distance between two points (x₁,y₁) and (x₂,y₂) is equal to the square root of the sum of the squares of the difference of the two provided points’ x and y coordinates. The formula for estimating the distance between two points is as follows. 

D = √(x₂−x₁)² + (y₂−y₁)²

How to Calculate the Distance Between a Point and a line :

To utilise the formula, the line’s equation must first be stated in standard form. A line’s standard form is Ax + By + C = 0. The point’s coordinates must also be known (x₁,y₁). The formula for calculating the distance between two points is:

distance = | Ax₁ + By₁ + C |/√(A² + B²)

Here’s how to utilise the point-to-line distance formula:

• Write the line’s equation in standard form: Ax + By + C = 0.

• Determine A, B, C, as well as x1 and y1.

• Enter the figures into the formula and solve.

• The distance between the point and the line is the solution.

Distance Formula: 

Any distance formula, as the name implies, provides the distance (the length of the line segment). The distance between two points, for example, is the length of the line segment joining them. The Pythagorean theorem is used to obtain the formula for distance between two points in a two-dimensional plane, which may also be extended to estimate distance between two points in a three-dimensional plane. In coordinate geometry, there are several forms of distance formulae.

Let us take an example to understand the distance between two points,

1) Using the distance of a point from a line formula, calculate the distance from point K(3,7) to line PQ y=(6/5) x + 2.

Solution: Let us first express the provided line in standard form. 

The line PQ may be reduced as follows:

y=(6/5) x + 2

5y = 6x +10

Hence, 6x – 5y + 10 = 0

d = | Ax₁ + By₁ + C |/√(A² + B²) is the formula for the distance of a point from a line.

The coordinates of the point K are given here is K(x₁,y₁) = (-3, 7), and A = 6, B =-5 and C = 10

d = |(6)(-3) + (-5)(7) + 10|/ √((6)²+(-5)²)

= |-18 -35 + 10|/ √(36 + 25)

= |-43|/√(61)

= |-5.506|

So the perpendicular distance between K (-3, 7) and the line PQ 6x – 5y + 10 = 0 is 5.506 units.

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 nearest point on the line is the smallest distance from that point, which is perpendicular to the line.