A Brief Study on Circles

A circle is a collection of all points that are equidistant from a given point. The centre of the circle is the point from which all the points on a circle are equidistant, and the radius of the circle is the distance from that point to the circle. The centre of a circle is designated by a single letter.

The given circle has a radius of r and a centre at C. Because all points on a circle are the same distance from the centre and the radii of a circle have one endpoint on the circle and one at the centre, all radii of a circle are congruent by definition.

Every circle has a diameter as well. A circle’s diameter is the segment that contains the centre and has both endpoints on the circle. As a result, all circle diameters are equivalent.

A circle is another common type of simple closed curve. Circles are geometric forms in which all of the points are at the same distance from the centre. Because they are not formed up of segments, they are not polygons. Points on the same line, such as those in a segment, are never equidistant (at the same distance) from each other.

Because circles are unlike any other geometric form, they have their own set of geometric laws. These guidelines will be put down in the subsequent classes, but not expounded upon. We’ll lay the framework for analysing angles both within and outside of a circle by defining certain characteristics of circles such as arcs, chords, diameters, radii, and central angles. Then we’ll talk about geometric shapes like tangent lines and secant lines, which are mostly outside of a circle.

The following lessons are an attempt to present some of the basic ideas related to circles; they are not a comprehensive examination of circles’ importance in geometry. Definitions and a few key traits will be provided in these sessions. We’ll focus more on solving for unknown parts in the Geometry 2 SparkNotes, and we’ll go through the features of circles and their associated geometric figures in depth. Contrary to common assumption, circles do occur regularly in nature. The majority of rotational scenarios include circles and/or circular movement. Reeling in a fishing line, driving a vehicle on wheels, and the Earth rotating are all examples of rotation scenarios.

What are the geometry properties of circles?

A circle is a two-dimensional form made up of a succession of points in the plane that are at a fixed or constant distance (radius) from a fixed point (centre). The origin or centre of the circle is the fixed point, while the radius is the fixed distance between the points. Every point on the circle is an equal distance from the circle’s centre. A circle is a two-dimensional shape with a defined radius.

In mathematics, there are several properties of circles that focus on geometry. It may also be displayed in relation to straight lines, polygons, and angles. When considered together, all of these features are properties of the circle. Now that we have a clearer knowledge of what a circle is, let’s look at some of its essential characteristics. To discover more, keep reading.

Properties of circle related to chord

Theorem 1: The chord is bisected by a perpendicular from the circle’s centre to the chord.

A chord with the centre O.AB is a circle with the centre OM⊥AB

To prove:AM=BM

Proof: In∆AOM & ∆BOM

AO=OB  (Radius)

OM=OM (COMMON)

∠AMO=∠BMO=90°

∴△AOM≅△BOM (By RHS congruency rule)

∴AM=BM (By CPCT)

Theorem 2: The perpendicular line segment connecting the circle’s centre and the chord’s midpoints.

AB is a chord such that OM⊥AB.

To prove: AM=BM

Proof: In ∆AOM & ∆BOM

AO=OB (Radius)

OM=OM (Common)

AM=AM (Given)

∴∆AOM≅∆BOM (By SSS rule)

∴∠AMO=∠BMO=90°(By CPCT)

What are the rules of circle in geometry?

In mathematics, the circle is the most intriguing shape. The circle is a locus that connects any locations that are the same distance apart.

In circular geometry, there are twelve rules.

1. At the centre, equal arcs/chords subtend equal angles. 

2. Equal chords are separated by the same distance from the centre. Equal distance from the centre equals equal chords.

3. The chord is perpendicular to the line connecting the circle’s centre to the chord’s centre. The chord is bisected by a perpendicular line from the chord to the centre.

4. Standing on the same chord/arc, the angle at the centre is double that.

5. On the same arc, all angles are equal.

6. A semi-circle has angles 90°.

7. Cyclic quadrilaterals are quadrilaterals having all four corners on a circle. The opposing angles are the supplementary angles of cyclic quadrilaterals.

8. The radius and the tangent form a 90° angle.

9. The length of all tangents taken from an exterior point is the same.

10. When two chords or secants (AB and CD) cross at X, AX.AX=CX.DX.

11. A secant (ABX) and a tangent (CX) cross at X, then AX.BX=CX∧2

12. The alternate segment’s chord and tangent produce the same angle.

Conclusion

A circle is a planar shape produced by a closed curve whose points are all the same distance apart from the circle’s centre. A circle is a planar shape produced by a closed curve whose points are all the same distance apart from the circle’s centre. Everyone uses the notion of circles in geometry on a daily basis. It’s employed in agriculture, education, and architecture.