Circle geometry

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A circle is a closed curve formed by a set of points on a plane that are at the same distance from its center O. That distance is known as the radius of the circle.

 

Primary definitions

 

Circle and major definitions

 

Arc: An arc is any line segment which is part of the circumference.

Diameter: The diameter of a circle is a line segment that passes through the center of the circle and has its endpoints on the circle. All the diameters of the same circle have the same length.

Radius: The radius of a circle is a line segment from the center of the circle to a point on the circle.

In the above figure, O is the centre of the circle, OP is the radius. The radii of a circle are of the same length and are half of the length of the diameter.If the radius is r, then,

Area of a circle = πr2

Circumference of a circle = 2πr

From the above diagram, the length of the arc AOB is given by 

The area of the arc AOB is given by

Tangent: A tangent is a line that touches a circle at only one point. A tangent is perpendicular to the radius at the point of contact.

Chord: A chord is a line segment with both endpoints on the circle, but it may not pass through the center of the circle.

Secant: it is a line which intersects the circle at two distinct points.

 

General Properties of Circles

 

The angle subtended by the diameter is 90 degrees. A perpendicular dropped on a chord from the centre of a circle bisects the chord.

 

1. Angle subtended by diameter

The angle subtended by the diameter in a semi-circle is a right angle.

2. Angle on circumference and centre

The measure of an inscribed angle is half the measure of the angle made by its intercepted arc at the centre.

3. Quadrilaterals

Opposite angles in a cyclic quadrilateral add up to 180o.

 

Chords of a circle

1. Perpendicular bisector of chords

The perpendicular from the centre of a circle to a chord of the circle bisects the chord. Conversely, the line joining the centre of the circle and the midpoints of the chord is perpendicular to the chord.

2. Distance from centre

Equal chords of a circle or congruent circles are equidistant from the center. Conversely, two chords of a circle or of congruent circles that are equidistant from the center and are making the same angle with the line drawn from the centre of the circle are equal.

3. Angle subtended on circumference

In a circle, equal chords(and arcs) subtend equal angles on any point of the circumference. Conversely, if a line segment joining two points subtends equal angles at the two other points lying on the same side of the line containing the segment, then the four points lie on the same circle.

4. Angle subtended at centre

In a circle, equal chords(and arcs) subtend equal angles at the centre. Conversely, chords which subtend equal angles at the centre of the circle are equal.

 

Tangents

Common Tangents

 

1. Angle with radius

The tangent at any point of a circle and the radius through that point are perpendicular to each other.

2. Length of tangents

The lengths of two tangent segments, from the exterior point to the circle are equal.

3. Alternate segment rule

The angle that the tangent to the circle makes with a chord drawn from a point of contact is equal to the angle subtended by that chord in the alternate segment of the circle.

4. Common tangents

For the two circles, AP and DS are the direct common tangents and BQ and CR are the transverse common tangents, where r1 and r2 are the radii of the two circles.

 


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