Perpendicular Lines in Geometry

To understand the concept of perpendicular lines let’s start from the beginning. Do you know what a line is and what is it made up of? Yes, you got that right. A line consists of infinitely small points or dots. The meaning of a line is simple. A line is a one - dimensional figure which joins infinite points on a graph and has no starting or ending point. There are 2 types of intersecting lines and parallel lines. Intersecting lines are when two lines meet each other at one common point. Parallel lines are two lines which never meet each other and are at a constant distance from each other.

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Some Common Sense Concepts of Perpendicular Lines

For a better understanding, let’s have some perpendicular line examples. We all know what a square is but have you ever noticed that a rectangle is made from the intersection of two sets of perpendicular lines. Similarly, when the adjacent sides of this rectangle are equal it makes a square. The sides of the right-angled triangle which encloses the 90° also made perpendicular lines. Did you know that the diagonals of the rhombus are perpendicular too? Also, when two lines are perpendicular to the same line, they are paralleling to each other and shall never intersect. 

Representation of Perpendicular lines

In mathematics, perpendicular lines represented by a symbol ‘⊥’.

For example, there are two lines ‘L’ and ‘M’. To express this, we’ll write L⊥M which is read as L is perpendicular to M. To find the shortest between any two objects we firstly draw a perpendicular line segment which joins the objects. The length of this line segment is the answer. Perpendicular lines are such an important concept of geometry that without it, we would not know the concept of shapes, bases of the theorem, and the entire field of trigonometry. ‘Perpendicular lines’ are man’s biggest discovery. Can you imagine that all the possible shape and structure can be drawn between those perpendicular lines of the graph?

Well, every possible line drawn in this world has a slope. Perpendicular lines are no different. Before we jump into the slope of the perpendicular line first let’s know about slopes. The slope of a line tells us how fast it is rising or falling. If expressed in mathematical terms, the slope of a line is the change in a line's y-value concerning the change in the line's x-value. When it comes to two perpendicular lines, the slopes are negative reciprocals of each other. This means that if a line is perpendicular to a line that has a slope l, then the slope of the line is -1 / l. Right angles formed by two intersected lines are drawn the product of their slopes equals −1. 

Perpendicular lines in Geometrical Objects

The heights of a triangle are perpendicular to their respective bases in a right-angled triangle. The perpendicular bisectors of the sides also play an important role in triangle geometry.

An isosceles triangle has a line which passes the centroid, orthocenter, circumcenter, and centre of the nine - point circle is perpendicular to the triangle's base. This line is also known as the Euler line.

In a square or a rectangle, all the adjacent sides form perpendicular lines. A right trapezoid has two pairs of adjacent sides that are perpendicular. Each of the four altitudes of a quadrilateral is perpendicular to a side through the midpoint of the opposite side. An orthodiagonal quadrilateral is a figure whose diagonals are perpendicular. By Brahmagupta's theorem, in an orthodiagonal quadrilateral which is also cyclic, a line through the midpoint of one side, and the intersection point of the diagonals is perpendicular to the opposite side.

Every diameter of a circle is perpendicular to the tangent line to that circle at the point where the diameter intersects the circle. A line segment that touches both ends of a circle is called a chord and when a line segment is drawn through the circle's centre bisects the chord and is perpendicular to the chord.

The Thales' theorem states that when two lines pass through the same point on a circle but passing through the opposite endpoints of a diameter are perpendicular. This is like saying that any diameter of a circle subtends a right angle at any point on the circle, except the 2 endpoints of the diameter.

The major and minor axes of an ellipse also form perpendicular lines. It also is perpendicular to the tangent lines to the ellipse at the points where the axes meet the ellipse. The major axis of an ellipse is perpendicular to the directrix and every latus rectum.

In a parabola, the axis of symmetry is perpendicular to every latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola. 

Perpendicular is the best example to give whenever we talk about beauty in simplicity. It is the basis of every math concept we are aware of. A perpendicular line is something that is so easy yet so complicated and relevant in every aspect of life which we tend to ignore.