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.

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:

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:

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:

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

In theTutorelablog, 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

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

Video Solution

Answer

Exercise #2

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

Video Solution

Answer

Exercise #3

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

Video Solution

Answer

8

Exercise #4

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

Video Solution

Answer

2

Exercise #5

Calculate the length of the arc marked in red given that the circumference is 6.

Video Solution

Answer

$\frac{5}{6}$

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