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Introduction: 

According to mathematics, A circle is a two-dimensional object in which all points on the circle’s surface are equidistant from the center point. The segment of the circle can be defined as the area between the chord and the arc of the corresponding circle. The circle is the subject of several theorems. 

A circle segment is defined as the area enclosed by an arc and a chord of the circle. Let’s review the definitions of an arc and a circle chord. An arc can be defined as a section of the circumference of a circle. A chord is a section of line that connects any two locations on the circumference of a circle.

A minor segment and a major segment are the two types of segments that exist. A minor arc of the circle creates a minor segment, and a major arc of the circle creates a major segment. Let R be the radius of the arc that forms part of the segment’s perimeter, an area of the segment, c the chord length, s the arc length, h the height of the segment, and θ the central angle subtending the arc in radians.

The chord length and height, as well as the arc length as part of the perimeter, are usually given or measured, while the unknowns are usually the area and sometimes the arc length. Because the radius and central angle cannot be determined directly from chord length and height, two intermediary values, the radius and central angle are commonly calculated first. 

Alternate Segment Theorem:

According to this theorem, The angle created by the tangent and the chord across one of the endpoints is equal to the measure of the angle opposite in the alternate segment.

When a chord is drawn through the point of contact of a tangent to a circle, the angles created by the chord with the given tangent are identical to the angles formed by the corresponding alternate segments.

Assume that a tangent is formed to a circle with P as the point of contact and that a chord PQ is drawn through P at an angle α to the tangent.

Assume that PQ subtends an angle β at any point R along the circle’s circumference, as shown:

The alternate angles in the alternate segment for the angle between the tangent at P and the chord PQ are ∠PRQ = ∠β

 

Regardless of R’s position, this angle will remain constant (as long as R stays in the segment opposite the tangent).

Proof of the Alternate Segment Theorem

The angle formed by the chord and the tangent in the alternate segment is identical to the angle formed by the chord in the first segment.

 

Let’s put this to the test.

Let P be a point on the circle’s circumference. Assume the role of the circle’s center.

Point to Remember:

  • One of the circle theorems is the alternate segment theorem. “For any circle, the angle formed by the chord in the alternate segment is equal to the angle formed by the tangent and the chord across the point of contact of the tangent,” says the theorem. The tangent-chord theorem is another name for the alternate segment theorem.
  • The angle formed by the chord and the tangent is identical to the angle formed by the chord in the other segment, according to the alternate segment theorem.
  • This theorem aids us in determining the unknown angles of any polygon encircled by a circle.
  • The angle measured between a circle chord and a tangent through any of the chord’s endpoints is the same as the angle measured in the alternate segment.
  • A tangent drawn to a circle and a chord drawn from the same point is known as the Alternate Segment Theorem. The angles in the alternate