PERPENDICULAR LINE: CHARACTERISTICS, EXAMPLES, EXERCISES - MATHS - 2025

A perpendicular line is one that forms an angle of 90º with respect to another line, curve or surface. Note that when two lines are perpendicular and lie on the same plane, when they intersect, they form four identical angles, each 90º.

If one of the angles is not 90º, the lines are said to be oblique. Perpendicular lines are common in design, architecture and construction, for example the pipe network in the following image.

Figure 1. Network of pipes at right angles and numerous perpendicular lines. How many 90º angles can be counted in this image? Source: Piqsels.

The orientation of the perpendicular lines can be diverse, such as the ones shown below:

Figure 2. Perpendicular lines on the plane. Source: F. Zapata.

Regardless of position, lines perpendicular to each other are recognized by identifying the angle between them as 90 °, with the help of the protractor.

Note that unlike parallel lines in the plane, which never intersect, perpendicular lines always do so at a point P, called the foot of one of the lines on the other. Therefore two perpendicular lines are also secant.

Any line has infinite perpendiculars to it, since just by moving segment AB to the left or right on segment CD, we will have new perpendiculars with another foot.

However, the perpendicular that passes just through the midpoint of a segment is called the bisector of that segment.

Examples of perpendicular lines

Perpendicular lines are common in the urban landscape. In the following image (figure 3) only a few of the many perpendicular lines that can be seen in the simple facade of this building and its elements such as doors, ducts, steps and more have been highlighted:

Figure 3. There are a large number of perpendicular lines on the facade of a common building like this. Source: Richard Kang via Flickr.

The good thing is that three lines perpendicular to each other help us to establish the location of points and objects in space. They are the coordinate axes identified as the x-axis, y-axis, and z-axis, clearly visible in the corner of a rectangular room like the one below:

Figure 4. The Cartesian axis system consists of three lines perpendicular to each other, each one has a preferential direction in space. Left Image Credits: treybunn 2 via Flickr. Right image; Needpix.

In the panorama of the city, on the right, the perpendicularity between the skyscraper and the ground is also noticed. The first one we would say is along the z axis, while the ground is a plane, which in this case is the xy plane.

If the ground constitutes the xy plane, the skyscraper is also perpendicular to any avenue or street, which guarantees its stability, since an inclined structure is unstable.

And in the streets, wherever there are rectangular corners, there are perpendicular lines. Many avenues and streets have a perpendicular layout, as long as the terrain and geographical features allow it.

To express abbreviated perpendicularity between lines, segments or vectors, the symbol ⊥ is used. For example, if line L ₁ is perpendicular to line L ₂, we write:

L ₁ ⊥ L ₂

More examples of perpendicular lines

- In the design the perpendicular lines are very present, since many common objects are based on squares and rectangles. These quadrilaterals are characterized by having internal angles of 90º, because their sides are parallel two by two:

Figure 5. Squares and rectangles are part of many designs, such as this simple cardboard box to store merchandise. Source: F. Zapata.

- The fields in which different sports are practiced are demarcated by numerous squares and rectangles. These in turn contain perpendicular lines.

- Two of the segments that make up a right triangle are perpendicular to each other. These are called the legs, while the remaining line is called the hypotenuse.

- The lines of the electric field vector are perpendicular to the surface of a conductor in electrostatic equilibrium.

- For a charged conductor, equipotential lines and surfaces are always perpendicular to those of the electric field.

- In piping or conduit systems used to transport different kinds of fluids, such as gas that appear in figure 1, it is common to have right-angle elbows. Therefore they form perpendicular lines, such is the case of a boiler room:

Figure 6. Pipes in a boiler room. Source: Wikimedia Commons. Roger McLassus / CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)

Exercises

- Exercise 1

Draw two perpendicular lines using a ruler and a compass.

Solution

It is very simple to do, following these steps:

-The first line is drawn, called AB (black).

-Above (or below if you prefer) AB mark point P, through which the perpendicular will pass. If P is just above (or below) the middle of AB, that perpendicular is the bisector of segment AB.

-With the compass centered on P, draw a circle that cuts AB at two points, called A 'and B' (red).

-The compass is opened at A'P, it is centered on A 'and a circumference is drawn that passes through P (green).

-Repeat the previous step, but now opening the measure the length of the segment B'P (green). Both arcs of circumference intersect at point Q below P and of course at the latter.

-The points P and Q are joined with the ruler and the perpendicular line (blue) is ready.

-Finally, all auxiliary constructions must be carefully erased, leaving only the perpendicular ones.

Figure 6. Tracing of perpendicular lines with a ruler and compass. Source: Wikimedia Commons.

- Exercise 2

Two lines L ₁ and L ₂ are perpendicular if their respective slopes m ₁ and m ₂ meet this relationship:

m ₁ = -1 / m ₂

Given the line y = 5x - 2, find a line perpendicular to it and that passes through the point (-1, 3).

Solution

-First is the slope of the perpendicular line m _⊥, as indicated in the statement. The slope of the original line is m = 5, the coefficient that accompanies "x". So:

m _⊥ = -1/5

-Then the equation of the perpendicular line y _{⊥ is constructed,} substituting the previously found value:

y _⊥ = -1 / 5x + b

-Next, the value of b is determined, with the help of the point given by the statement, the (-1,3), since the perpendicular line must pass through it:

y = 3

x = -1

Substituting:

3 = -1/5 (-1) + b

Solve for the value of b:

b = 3- (1/5) = 14/5

-Finally, the final equation is built:

and _⊥ = -1 / 5x + 14/5

References

1. Baldor, A. 2004. Plane and space geometry. Cultural Publications.
2. Clemens, S. 2001. Geometry with applications and problem solving. Addison Wesley.
3. Math is Fun. Perpendicular lines. Recovered from: mathisfun.com.
4. Monterey Institute. Perpendicular lines. Recovered from: montereyinstitute.org.
5. Wikipedia. Perpendicular lines. Recovered from: es.wikipedia.org.