Perpendicular straight lines


Perpendicular lines are those that when they cross form four equal angles, each one being a right angle, that is, measuring 90º.

Seen in another way, when two perperndicular lines intersect, a complete or perigonal angle is divided into four identical parts.

Perpendicular lines are a possibility among the cases of secant lines. These are those that intersect or, to put it another way, have a point in common.

It is worth remembering that a straight line is an indefinite sequence that goes in only one direction, that is, it does not present curves, and it has neither a beginning nor an end.

Equation of perpendicular lines

If line 1 and line 2 are perpendicular, the slope of one is equal to the inverse of the slope of the other and with the sign changed from positive to negative or vice versa. That is, if on line 1 the slope is, for example, 1/5, on line 2, the slope will be -5. Seen another way, it is true that:

m1 = -1 / m2

In the equation, m1 is the slope of line 1, while m2 is the slope of line 2, both of which are perpendicular.

Let us remember that, in analytic geometry, a line can be represented by an equation of the following type:

y = mx + b

Thus, in the equation y is the coordinate on the ordinate axis (vertical), x is the coordinate on the abscissa axis (horizontal), m is the slope (inclination) that forms the line with respect to the abscissa axis , and b is the point where the line intersects the ordinate axis.

We can see in the image below that the slope of one of the lines is -2, and that of the other, 0.5, which is the same as 1/2. In this way, what is explained above is fulfilled.

Example of perpendicular lines

We can determine if two lines are perpendicular by knowing two of their points. For example, suppose line 1 passes through point A (0.5,4) and point B (0, 2). Meanwhile, line 2 passes through point C (2, 2.5) and point D (-2, 3.5). Are line 1 and line 2 perpendicular?

First, we find the slope of line 1, dividing the variation on the y-axis by the variation on the y-axis when we go from point A to point B. Thus, on the y-axis we go from 4 to 2, varying by -2. Meanwhile, on the x-axis, we go from 0.5 to 0, varying by -0.5. Therefore, m1 being the slope of line 1:

m1 = (2-4) / (0-0.5) = - 2 / -0.5 = 4

Then we find the slope of line 2 (m2). We proceed in the same way, but going from point C to point D.

m2 = (3.5-2.5) / (- 2-2) = 1 / (- 4) = - 1/4 = -0.25

As we see, m1 = -1 / m2 since 4 = - (1 / -0.25). Therefore, line 1 and line 2 are perpendicular.

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