Chord of a Circle Definitions and Examples
In geometry, a chord is a line segment that connects two points on a curve or circle. A circle can have any number of chords, but some are more important than others. The length of a chord is the distance between the two points that it connects. The midpoint of a chord is the point on the circle that is equidistant from the endpoints of the chord. Chords are used in a variety of mathematical applications and are a crucial part of understanding circles. In this blog post, we will explore the definition of chords and provide examples to illustrate their importance.
Chord of Circle
A circle’s chord is a line segment that connects any two points on the circle. The word “chord” is from the Latin corda, meaning “rope,” and it’s an apt term for this geometric figure.
A cord can be horizontal, vertical, or diagonal. It can also be a straighter line or a curve. The important thing to remember is that a cord always passes through the center of the circle.
The length of a cord is the distance between the two points that it connects. To find the length of a cord, you can use the Pythagorean theorem if you know the coordinates of the points. For example, if one point has coordinates (2, 3) and the other has coordinates (6, 7), then the length of the cord is:
?((6-2)² + (7-3)²) = ?(4² + 4²) = ?32 = 5.66 units
You can also use trigonometry to find the length of a cord if you know the angle that it forms with respect to the circle’s diameter. For example, if a cord forms a 30° angle with the diameter of a circle, then you can use soh cah toa to find that:
cord length = diameter * sin(30°) = 10 * sin(30°) ? 8.66 units
What is the Chord of a Circle?
A circle is a simple closed curve and its chord is a line segment joining any two points on the curve. The word chord is derived from the Latin word chorda, which means “string.” In plane geometry, a circle’s chords are line segments that intersect the circle at two points. A central angle is an angle whose vertex is at the center of a circle and whose sides intercept arcs of the circle. The intercepted arc creates an inscribed angle, which is equal to half the central angle.
Properties of the Chord of a Circle
A chord of a circle is a straight line segment whose endpoints both lie on the circle. A chord can be defined by its two endpoints, or by its midpoint and one endpoint. The length of a chord is the distance between its two endpoints.
The following properties hold for any chord of a circle:
* The length of the chord is always less than or equal to the diameter of the circle.
* The longer the chord, the closer it is to being perpendicular to the radius at its midpoint. In other words, chords that are close to being perpendicular to the radius have longer lengths.
* Chords that are parallel to each other have equal lengths.
Formula of Chord of Circle
A chord of a circle is a straight line that joins two points on the circumference of the circle. The length of the chord is the distance between the two points.
The formula for the chord of a circle is:
c = 2r sin(?/2)
where c is the length of the chord, r is the radius of the circle, and ? is the angle between the two points on the circumference.
For example, if you have a circle with a radius of 10 cm and you want to find the length of the chord when ? = 30°, you would use the formula above to calculate that c = 20 cm.
Theorems of Chord of a Circle
The theorems of chord of a circle are as follows:
1) The chords of a circle that pass through its center are equal in length.
2) Chords that intersect at right angles (90 degrees) inside a circle divide the chord into two segments that are also equal in length.
3) The perpendicular bisector of a chord passes through the center of the circle.
4) The diameter of a circle is always longer than any other chord.
5) The length of a minor arc is always less than the length of the corresponding major arc.
6) If two chords intersect inside a circle, then the product of their lengths is equal to the product of the lengths of the two segments into which they divide the other chord.
Theorem 2: Chords of a circle, equidistant from the center of the circle are equal.
If two chords of a circle are equidistant from the center of the circle, then those two chords are equal in length. This theorem is also sometimes called the “equal chords in a circle theorem” or the “chords of a circle theorem.”
Theorem 3: For two unequal chords of a circle, the larger chord will be closer to the center than the smaller chord. (Unequal Chords Theorem)
The Unequal Chords Theorem states that, for two chords of a circle that are not equal in length, the longer chord will be closer to the center of the circle than the shorter chord. This theorem can be applied in a variety of situations, such as when finding the length of a missing chord segment or determining the location of a point on a circle.
To prove this theorem, we will use the fact that the perpendicular bisector of a chord passes through the center of a circle. Consider two unequal chords AB and CD in a circle, with AB being the longer chord. We will construct the perpendicular bisector of each chord and show that it passes through the center O of the circle.
First, we construct the perpendicular bisector of AB. To do this, we find the midpoint M of AB and draw a line perpendicular to AB at M. Since M is the midpoint of AB, AM = MB. Therefore, angle OMB = angle OMA, since they are both right angles. Since angle OMB = angle OMA, line OM is perpendicular to line MN (the perpendicular bisector of AB). But we know that any line perpendicular to a chord must pass through the center of the circle, so OM passes through O.
Now we construct the perpendicular bisector of CD. We find the midpoint N of CD and draw a line perpendicular to CD at N. Since N is the midpoint of CD, AN = NC
The chord of a circle is a straight line that connects two points on the circumference of the circle
A chord is a straight line that connects two points on the circumference of a circle. The length of the chord is the distance between the two points.
The chords of a circle are important because they can be used to find the diameter of the circle. The diameter is the longest chord of a circle. It goes through the center of the circle and has equal lengths on each side of the center.
The length of the chord can be found using the Pythagorean theorem
The Pythagorean theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be used to find the length of the chord, as shown in the diagram below.
To find the length of chord AC, we first need to find the length of side BC. This can be done using the Pythagorean theorem, as follows:
BC2 = AB2 + AC2
BC = ?(AB2 + AC2)
Therefore, the length of chord AC is equal to ?(AB2 + AC2).
Chords are used to create arcs and sectors
Chords are created when two points on a circle are connected by a straight line segment, called the chord. Chords can be used to create arcs and sectors.
An arc is created when a chord is extended past one of the points it is connecting. The point at which the chord is extended is called the endpoint of the arc. The length of the arc can be measured in terms of degrees or radians, and it is equal to half the measure of the central angle formed by the two points on the circle.
A sector is created when a chord is extended past both of the points it is connecting. The area enclosed by a sector can be calculated using the formula A = r²(?/2), where r is the radius of the circle and ? is the measure of the central angle in radians.
A central angle is created when two chords intersect in the center of the circle
When two chords intersect in the center of a circle, they form a central angle. The central angle is the angle formed by the intersection of the two chords. The size of the central angle is determined by the amount of rotation required to bring one chord into alignment with the other.
The central angle can be used to calculate the length of the chords. If the length of one chord is known, then the length of the other chord can be determined by measuring the central angle and using the formula for calculating the length of a chord.
The central angle can also be used to determine the area of a circle. If the length of one chord is known, then the area of the circle can be calculated by measuring the central angle and using the formula for calculating the area of a circle.
The diameter of a circle is the longest chord
A chord is a line segment that connects two points on a circle. The diameter of a circle is the longest chord. It goes through the center of the circle and has equal lengths on each side of the center.
Chords are perpendicular to the tangent line at the point
A chord is a line segment that connects two points on a curve. A circle’s chords are perpendicular to the tangent line at the point where they intersect the circle. This property allows us to define chords in terms of their length and angle relative to the center of the circle.