9 junio, 2024

Perpendicular line: characteristics, examples, exercises

A perpendicular line It 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 one of 90º.

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

The orientation of the perpendicular lines can be different, such as those shown below:

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

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

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

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

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Examples of perpendicular lines

Perpendicular lines are frequent 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 façade of this building and its elements such as doors, ducts, steps and more have been highlighted:

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

In the panoramic view of the city, on the right, the perpendicularity between the skyscraper and the ground can also be seen. The first we would say is found along the z-axiswhile the ground is a plane, which in this case is the plane xy.

If the ground constitutes the plane xythe skyscraper is also perpendicular to any avenue or street, which guarantees its stability, since a leaning 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 briefly express the perpendicularity between lines, segments or vectors, the symbol ⊥ is used. For example, if line L1 is perpendicular to line L2, we write:

L1 ⊥ L2

More examples of perpendicular lines

– In the design, 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:

– The fields where 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 legswhile the remaining line is called hypotenuse.

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

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

– In the systems of pipes or ducts used to transport different kinds of fluids, such as those of gas that appear in figure 1, it is frequent that there are right-angled elbows. Therefore they form perpendicular lines, such is the case of a boiler room:

Exercises

– Exercise 1

Draw two perpendicular lines using ruler and compass.

Solution

It is very easy to do, following these steps:

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

-Above (or below if preferred) AB mark the 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 in A’P, it is centered in A’ and a circle is drawn that passes through P (green).

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

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

-Finally you have to carefully erase all the auxiliary constructions, leaving only the perpendicular ones.

– Exercise 2

Two lines L1 and L2 are perpendicular if their respective slopes m1 and m2 meet this relationship:

m1 = -1/m2

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

Solution

-First, find 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 built, 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

The value of b is cleared:

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

-Finally, the final equation is built:

y⊥= -1/5x + 14/5

References

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

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