Perpendicular to a chord from the center of a circle

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 $90°$ 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 $A$
Our chord will be blue and will be called $BC$ .
The vertical, which comes out from the center of the circle and will be perpendicular to the chord $BC$ .
We will mark it in red and call it $AD$ .

Detailed explanation

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Test yourself on the parts of a circle!

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

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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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Test your knowledge

Question 1

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

Question 2

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

Question 3

Is it correct to say:

'the circumference of a circle'?

1) Bisects the Chord

Let's see this in the illustration:

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

The perpendicular $AD$ cuts the chord $BD$ in half
So that:
$BD=DC$

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 $AD$ bisects the central angle
$∡CAB$
So that:
$∢A1=∢A2$

Do you know what the answer is?

Question 1

Where does a point need to be so that its distance from the center of the circle is the shortest?

Question 2

Which diagram shows the radius of a circle?

Question 3

Calculate the area of the section painted red given that the area of the circle is 12.

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 $AD$ crosses the arc in front of the chord.
So that:

$BE=EC$

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 $AD$ is perpendicular to $BC$
Then
$∢A1=∢A2$
$BD=DC$
$BE=EC$

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

Exercise #1

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is greater than the center of the circle will necessarily be outside the circle.

Answer

greater

Exercise #2

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

Video Solution

Step-by-Step Solution

Let's remember that the circular line draws the shape of the circle, and the inner part is called a disk.

Therefore, in diagram B, the point is located in the inner part, meaning inside the disk.

Answer

Exercise #3

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

Video Solution

Step-by-Step Solution

Let's remember that a radius is a line segment connecting the center of the circle to a point that lies on the circle itself.

In drawing A, the line doesn't touch any point on the circle itself.

In drawing B, the line doesn't pass through the center of the circle.

We can see that in drawing C, the line that extends from the center of the circle is indeed connected to a point on the circle itself.

Answer

Exercise #4

Where does a point need to be so that its distance from the center of the circle is the shortest?

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is less than the radius from the center of the circle will necessarily be inside the circle.

Answer

Inside

Exercise #5

Which diagram shows the radius of a circle?

Step-by-Step Solution

Let's remember that a radius is a line segment connecting the center of a circle to any point on the circle itself.

In drawing C we can see that the line coming from the center of the circle indeed connects to a point on the circle itself, while in the other drawings the lines don't touch any point on the circle.

Therefore, C is the correct drawing.

Answer

Check your understanding

Question 1

Calculate the length of the arc marked in red given that the circumference is 12.

Question 2

Calculate the length of the arc marked in red given that the circumference is 12.

Question 3

Calculate the length of the arc marked in red given that the circumference is 36.

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