Circle and Its Chords

A chord is a piece of a line that connects any two points on a circle. The line segment’s endpoints are on the circle’s perimeter. The Chord that passes through the centre of the circle is known as the Diameter. In a circle, it is the longest chord imaginable. The term “chord” comes from the Latin word “chorda,” which literally means “bowstring.” You’re already familiar with the terms arc and circumference. Let’s have a look at some theorems about circle chords.

Chord of a Circle Theorems

Theorem 1: At the centre of a circle, equal chords subtend equal angles.

Theorem 2: This theorem is the polar opposite of the previous one. It indicates that two chords are equivalent if they subtend equal angles at the centre.

Theorem 3: A perpendicular drawn from the circle’s centre to a chord cuts it in half. It indicates that the chords’ halves are of equal length.

Theorem 4: The clear line from the circle’s centre to the chords’ midpoint is perpendicular to it. To look at it another way, every line bisecting a chord from its centre is perpendicular to it.

Theorem 5: If three non-collinear points are present, then only one circle can cross through them.

Theorem 6: A circle’s equal chords are all equidistant from the centre.

Theorem 7: This theorem is the polar opposite of the previous one. It states that chords equidistant from a circle’s centre have the same length.

Theorem 8: An arc’s angle at the centre of a circle is double that of an arc’s angle at any other point on the circle.

Theorem 9: Angles created in the same circle segment are always identical in size.

Theorem 10: A line segment connecting two places is concyclic if it subtends equal angles at two other points on the same side. This indicates that they are all arranged in a circle.

Important Properties of Circle–Lines

The following are the properties of a circle.

1. Chord

A chord is a straight path with ends on the perimeter of a circle.

Properties of Chord

A chord is divided into two halves when a perpendicular is dropped from the centre.

2. Tangent

A tangent is a line that passes through the centre of the circle at any point.

Properties of Tangent

At the point where it touches the circle, the radius is always perpendicular to the tangent.

Important Properties of Circle – Related to Angles

Properties related to Angles in a circle

Inscribed Angle

The angle generated when two chords meet on the circle’s boundary is called an inscribed angle.

Inscribed Angles’ Characteristics

1. Angles created on the circumference of the circle by the same arc are always equal.

2. In a semi-circle, the angle is always 90 degrees.

Central Angle

A central angle is generated when two line segments meet in such a way that one of the line segments’ endpoints is at the circle’s centre and the other is at the circle’s perimeter.

Central Angles’ Property

The inscribed angle of an arc at its centre is twice the angle created by the same arc.

Important Circle Formulas: Area and Perimeter

The area and perimeter/circumference of a circle can be calculated using the following mathematical formulas.

Perimeter

• The perimeter or the Circumference of the circle is equal to 2 × π × R

• The length of an Arc is equal to (Central angle made by the arc/360°) × 2 × π × R

Area

• The area of the circle is equal to π × R²

• The area of the sector is equal to (Central angle made by the sector/360°) × π × R²

These are some of the fundamental theorems about circle chords and arcs.

Conclusion

The important properties of a circle related to chord, angles, tangent, and cyclic quadrilaterals have been provided in this section.

There are various features of circles in mathematics that focus on geometry. It can also be shown about straight lines, polygons, and angles. When considered together, all of these facts are characteristics of the circle.