Distance from a chord to the center of a circle

The distance from the chord to the center of the circle is defined as the length of the perpendicular from the center of the circle to the chord.
Theorems on the distance from the center of the circle:

Chords that are equal to each other are equidistant from the center of the circle.
If in a circle, the distance of a chord from the center of the circle is less than the distance of another chord from the center of the circle, we can determine that the chord with the lesser distance is longer than the other chord.

A1 - The distance from the chord to the center of the circle

All theorems can also exist in reverse.

Detailed explanation

Start practice

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.

Practice more now

Distance from the chord to the center of the circle

We already know what a chord in a circle is and of course we know what the center of the circle means.
But, have you thought about the distance between the center of the circle and the chord?
That's exactly why we're here! We will introduce you to the theorems about the distance of the chord from the center of the circle that you can use without needing to prove them.
The theorems about the distance of the chord from the center of the circle make sense, there's no reason not to understand them and remember them naturally.

First of all, it's important to know:

The distance of the chord from the center of the circle is defined for us as the length of the perpendicular line that goes from the center of the circle to the chord.
That is:

The distance is a vertical line that goes from the chord to the center of the circle

Shall we start?

The chords that are equal to each other are equidistant from the center of the circle.

Let's see this in the illustration:

We will mark $A$ as the center of the circle

If
$BC=DE$
Then
$AF=AG$

This theorem also works in reverse and therefore
if the distance of the chords from the center of the circle is equal, the chords are equal to each other.
Therefore:
If
$AF=AG$
Then
$BC=DE$

If in a circle, the distance of a chord from the center of the circle is less than the distance of another chord from the center of the circle, we can determine that the chord with the shorter distance is longer than the other chord.

This theorem can be a bit confusing.
To remember:
shorter distance - longer chord.
longer distance - shorter chord.

The truth is that you really don't have to memorize this sentence because you will see it beautifully in the illustration.
Let's see:

A3 - the chord with the shorter distance is longer than the other chord

In front of us is a circle.
We will mark - $A$ as the center of the circle.
We can clearly see that the chord $BC$ is shorter than the chord $ED$ .
We can also observe that for the shorter chord $BC$ , the distance is greater from the center of the circle
and rather to the longer chord $ED$ , shorter distance than the distance from the circle.
According to this theorem we can say that:
if
$AF<AG$
Then
$BD<ED$
This theorem also works in reverse, so we can also say that:
if
$BD<ED$
Then
$AF<AG$

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

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

Distance from the chord to 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.