Perpendicular to a chord from the center of a circle

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The perpendicular to the chord comes out of the center of the circle, intersecting the chord, the central angle in front of the chord and the arc in front of the chord.
Moreover, if there is a section that comes out from the center of the circle and crosses the chord, it will also be perpendicular to the chord.

We are here to present the properties of the perpendicular from the center of the circle to the chord.
First, we will remember that the perpendicular is a line that forms a 90° 90° degree angle.
Let's see it in the illustration:

A1 - Perpendicular to the chord from the center of the circle

In front of us, there is a circle.
We will mark the center of the circle with a letter AA
Our chord will be blue and will be called BCBC.
The vertical, which comes out from the center of the circle and will be perpendicular to the chord BCBC.
We will mark it in red and call it ADAD.

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Test yourself on the parts of a circle!

einstein

In which of the circles is the point marked in the circle and not on the circumference?

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Great! Now that we understand exactly what perpendicular from the center of the circle is, we'll move on to its characteristics.
You can use these characteristics without proving them.


Characteristics of the Perpendicular to the Chord from the Center of the Circle

The perpendicular from the chord, which comes out from the center of the circle:


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1) Bisects the Chord

Let's see this in the illustration:

A2 - Perpendicular to the chord from the center of the circle

The perpendicular ADAD cuts the chord BDBD in half
So that:
BD=DCBD=DC


2) Bisects the angle in front of the chord

Let's see this in the illustration:

A3 - Perpendicular to the chord from the center of the circle

The perpendicular ADAD bisects the central angle
CAB∡CAB
So that:
A1=A2∢A1=∢A2


Do you know what the answer is?

3) Bisects the arc that is in front of the chord

Let's see this in the illustration:

A4- Perpendicular to the chord from the center of the circle

The perpendicular ADAD crosses the arc in front of the chord.
So that:

BE=ECBE=EC


Please note:
Similarly, these characteristics also work in reverse, and we can say that if there is a straight line that comes out from the center of the circle and crosses the chord, then it is perpendicular to the chord.


In conclusion, a perpendicular line, coming from the center of the circle, crosses the chord, the relevant central angle, and the arc opposite the chord.
We will see all the features in an illustration:

A5 - Perpendicular to the chord from the center of the circle

If ADAD is perpendicular to BCBC
Then
A1=A2∢A1=∢A2
BD=DCBD=DC
BE=ECBE=EC


If you are interested in this article, you might also be interested in the following articles:

  • The center of the circle
  • Circle
  • Radius
  • Diameter
  • Pi
  • The circumference perimeter
  • Circular area
  • Arcs in a circle
  • Chords in a circle
  • Central angle in a circle
  • Inscribed angle in a circle
  • Distance from the chord to the center of the circle

In the Tutorela blog, you will find a variety of articles on mathematics.


Perpendicular to the chord from the center of the circle (Examples and exercises with solutions)

Exercise #1

In which of the circles is the point marked in the circle and not on the circumference?

Video Solution

Answer

Exercise #2

In which of the circles is the segment drawn the radius?

Video Solution

Answer

Exercise #3

How many times longer is the radius of the red circle than the radius of the blue circle?

220

Video Solution

Answer

5

Exercise #4

How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?

Video Solution

Answer

2

Exercise #5

How many times longer is the radius of the red circle than the radius of the blue circle?

168

Video Solution

Answer

2 2

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