Perpendicular

The shortest distance from a point to a line is the perpendicular distance from that point to the line. This is because any other path or line segment from the point to the line would be longer than the perpendicular.

Whether we want to draw a perpendicular line from a point on the line or from a point outside of it, we actually utilize the property of the isosceles triangle. I recommend taking a look at the topic of triangles.Go to section Side - Angle relationship.

Isosceles triangle

Assume you are standing upright on the ground, If you raise one arm higher than the other, the distance of the ray coming from your arm to the ground will be greater than the distance of the ray coming from your other arm to the ground.

If you extend your arms at equal angles, then the lengths of the rays coming from your arms to the ground will also be equal.

From another perspective, if we can obtain an isosceles triangle, it means we are standing upright. Not exactly, but almost. (We need two isosceles triangles.)

Constructing a perpendicular from a point outside a given line.

Let's try to form an isosceles triangle. For this, you need either a ruler or a compass.

Usually, a compass is used, but here I will first use a ruler, and then we will use a compass.

Create an isosceles triangle where the line and the point meet. I took the sides as 10 cm here, but you can create any isosceles triangle according to your preference.

I will draw a line from top to bottom, but we can't be sure which point it will intersect with, meaning whether it will be exactly 90 degrees or not. Therefore, let's draw another isosceles triangle, a reflection of the isosceles triangle, on the bottom side."

Now I can draw the line I want, as 'only one line passes through two points'. This line will be perpendicular to our original line.

We can repeat similar steps with a compass.

Place your compass on the point and draw two arcs in both directions that will intersect the line.

Mark the points where they intersect.

In fact, we have now obtained an isosceles triangle; look closely at the points.

Without changing the opening of your compass, place it on the points on the line and draw an arc towards the center.

Do the same step for the other point.

Connect the two points.

What actually happened? In fact, we've drawn two isosceles triangles again.

A compass is used to draw points that are equidistant from a given point. The distance from the black points on the arc to the black point where you placed your compass, and the distance from the red points on the arc to the red point where you placed your compass, are equal. The characteristic of the intersection point is that its distance is the same to both the red and black points (to the right and left). Therefore, it forms an isosceles triangle.

Drawing a perpendicular from a point on a given line.

If we can draw an isosceles triangle from this point, we will have drawn the perpendicular.

Why did we widen the angle of the compass in the third step? What kind of drawing would we get if we didn't widen it?

We did it in six steps; could we have finished our drawing in the fourth step?

Why is it important to be able to draw perpendicular to a line?

A perpendicular is a straight line or plane that is at a right angle (90 degrees) to another line or surface. In the context of geometry, when we say one line is perpendicular to another, we mean that the two lines intersect at a right angle.

Drawing perpendiculars is fundamental in geometry and various applications in real life. Here's why it's important:

Geometry and Constructions: Many geometric constructions, such as bisecting angles or drawing the altitude of a triangle, require drawing perpendiculars.

Engineering and Architecture: Ensuring right angles in designs and constructions is crucial for stability and aesthetics. For instance, the walls of a building are typically perpendicular to the ground.

Navigation and Mapping: Perpendicular lines are used to create grids, which help in plotting points and navigating.

Art: In perspective drawing, artists use perpendicular lines to ensure accurate proportions and angles.

Mathematical Proofs: In geometry, proving that lines are perpendicular can be essential to many proofs and theorems.

Everyday Life: Simple tasks like hanging a picture frame straight require an understanding of perpendicularity.

In essence, the ability to draw and recognize perpendicular lines is a foundational skill in many disciplines and everyday tasks.