Perpendicular Length Calculator (Distance)

Perpendicular Length Calculator

Perpendicular length calculator is used to calculate the perpendicular distance of a line from a point or coordinates.

What is Perpendicular Length?

The perpendicular length refers to the shortest distance between a point and a line or between two intersecting lines that form a right angle. It is a measurement that describes the vertical distance or height from a reference point to a line or plane.

The perpendicular length is always perpendicular to the line or plane it is measured against.

Properties of Perpendicular Length:

Here are some properties of the perpendicular length.

Perpendicularity:

The line connecting a point to another line or the line intersecting another line at a right angle forms a perpendicular length.

Shortest Distance:

Perpendicular length represents the shortest possible distance between the point and the line.

Right Angle:

The perpendicular length forms a right angle with the line or plane it is measured against.

Unique Solution:

For a given point and line, there is only one perpendicular length.

The formula of Perpendicular length:

The perpendicular length (PL) can be calculated using the following formula:

PL = |Ax + By + C| / √ (A² + B²)

Where:

• A, B, and C are the coefficients of the equation of the line.
• (x, y) represents the coordinates of the point from which the perpendicular length is measured.
• | | denotes the absolute value or magnitude of the expression.
• √ denotes the square root.

This formula allows us to determine the perpendicular length between a point and a line by considering the equation of the line and the coordinates of the point.

Applications of Perpendicular Length:

The term “Perpendicular length” helps us a lot in different aspects, here are some real-life applications of the perpendicular length.

Perpendicular Length in Architecture:

In architecture, perpendicular length plays a vital role in determining the dimensions and alignment of structures. It helps architects in creating precise layouts, measuring heights, and ensuring verticality in columns, walls, and beams.

Perpendicular Length in Engineering:

Engineers utilize perpendicular length measurements for designing structures, such as bridges, buildings, and mechanical components. It aids in maintaining accuracy and stability in construction, ensuring that the components are aligned properly.

Perpendicular Length in Mathematics:

Perpendicular length is a key element in geometric calculations. It allows mathematicians to solve problems related to angles, triangles, and polygons. The concept of perpendicular length is foundational in trigonometry, where it is used to determine side lengths and angles in right-angled triangles.

Perpendicular Length in Physics:

In physics, perpendicular length is crucial in understanding forces and vectors. It helps determine the vertical component of a force acting on an object or the displacement of an object in relation to a reference point.

Perpendicular Length in Surveying:

Surveyors rely on perpendicular length measurements to map out land, establish boundaries, and create accurate topographic representations. Perpendicular measurements assist in determining elevations, contours, and slope angles.

How to find perpendicular length?

Example 1:

Consider a line with the equation 2x + 3y - 6 = 0. Find the perpendicular length from the point P(4, 1) to this line.

Solution:

Step 1: Extract the values

A = 2, B = 3, C = -6 x = 4, y = 1

Step 2: Calculations

Using the formula:

PL = |Ax + By + C| / √ (A² + B²)

PL = |2(4) + 3(1) - 6| / √ (2² + 3²)

PL = |8 + 3 - 6| / √ (4 + 9)

PL = |5| / √13

PL = 5 / √13

Therefore, the perpendicular length from point P (4, 1) to the line 2x + 3y - 6 = 0 is 5 / √13.

Example 2:

Consider a line with the equation -3x + 2y + 9 = 0. Find the perpendicular length from the point Q(-1, 4) to this line.

Solution:

Step 1: Extract the values

A = -3, B = 2, C = 9 x = -1, y = 4

Step 2: Calculations

Using the formula:

PL = |Ax + By + C| / √ (A² + B²)

PL = |-3(-1) + 2(4) + 9| / √ ((-3)² + 2²)

PL = |3 + 8 + 9| / √ (9 + 4)

PL = |20| / √13

PL = 20 / √13

Therefore, the perpendicular length from point Q (-1, 4) to the line -3x + 2y + 9 = 0 is 20 / √13.