What is a real life example of a chord?

You have a cookie that you have to cut into two parts to share with your brother. If you cut the cookie with a straight line from one edge of the cookie to another, that line is a chord. It doesn't matter if one piece is huge and the other piece is tiny, or if they are exactly equal; the line you have cut is a chord.

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How to Draw a Chord

To draw a chord, one needs to define two points over the same circumference.

Two points of a circumference.

Now draw a line that connects the two points to each other. This line is the chord.

Chord: a line connecting two points on a circumference.

Theorems of Chords in the Circle

There are a number of theorems that involve the chord of a circle. The following images enumerate some components that are important to understanding each theorem.

Components of a circle.

- The segment connecting points A and B is the chord.

- The segment connecting the center of the circle to point C is called the radius. The radius measures half of the diameter of a circle. It connects the center to any point of the circumference. - The segment connecting point E to the center is called the apothem. It goes out from the center of the circle and divides a chord into two equal parts. The apothem is always perpendicular (meaning that it makes a 90 degree angle) to the chord. - The segment connecting points E and F is called the Sagitta. It connects the point that bisects a chord perpendicularly to a point on the circumference. The following image contains a component of the circle called a circular sector, which looks like a pizza slice.

Circular sector.

The image presents two circular sectors: the major (white) and minor (red) sectors. These two sectors are delimited by two radii (AC and AB), and an arc (BC). The major sector is delimited by the longer arc that connects B and C, and the minor sector is delimited by the shorter arc. The sum of the angles of the two sectors is always equals to 360 degrees due to the nature of the circle. The next image presents a component of the circle called a circular segment.

Circular segment.

Circular segments (green) are a part of a circle delimited by an arc (smaller than 180 degrees) and a chord (like the segment CB in the image). Based on the aforementioned properties of a circle, the theorems below can now be understood: Theorem 1: A segment that is perpendicular to a chord — and which is drawn from the center of a circle — bisects the chord (see apothem).

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Theorem 2: Two chords whose bisection points are at the same distance from the center of a circle are equal. Theorem 3: Considering two different chords in a circle, the chord that is closer to the center is the longer of the two chords.

Notice how chord AC is longer than chord DE.

Calculating the Length of the Chord

The following section presents two methods to calculate the length of circular chords. Method 1: Finding the length of a chord when the radius and central angle are known.

The formula to calculate the chord using this method is:

{eq}C = 2 * R * sin (frac {Theta}{2}) {/eq}

Where:

- C is the length of the chord.

- R is the radius of the circle.

- {eq}Theta {/eq} is the central angle.

Method 2: Finding the length of a chord when the radius of the circle and the distance between its center and the chord (apothem) are known.

Using method 2, chord length can be calculated by following this formula:

{eq}C = 2 * sqrt {(R^2 - d^2)} {/eq}

Where:

- C is the length of the chord.

- R is the radius of the circle.

- d is the length of the apothem of the circle.

Example 1: Find the length of the chord of a circle with a 2 meter radius. The central angle of the chord is {eq}60^circ {/eq}.

Solution: If R = 2 m, and Theta = {eq}60^circ {/eq}, then

{eq}C = 2 * R * sin (frac {Theta}{2}) {/eq}

becomes

{eq}C = 2 * 2 * sin (frac {60^circ}{2}) C = 4 * sin (30^circ) C = 4 * 0.5 = 2 meters. {/eq} Example 2: Find the length of the chord of a circle with a 2 meter radius and an apothem equal to 0.5 m.