Circle and Its Properties

 

Circle and Its Properties

 

CIRCLE

A circle is a set of points which are equidistant from a given point. The given point is known as the centre of that circle.

 

Arcs, Chords and Central Angles

·         In equal circles (or in the same circle), if two arcs are equal, the chords associated with the arcs are also equal and vice-versa.

·         In equal circles (or in the same circle), if two arcs subtend equal angles at the centre, then they are equal.

 

Angles in a Circle

·         The angle which is subtended at the centre by an arc of a circle is double the angel which is subtended at any point on the remaining part of the circle.

·         Angles in the same segment of a circle are equal.

·         The angle in a semi-circle is a right angle.

 

PARTS OF CIRCLE AND THEIR PROPERTIES

Chords in a Circle

·         A straight line drawn from the centre of a circle to bisect a chord which is not a diameter is at right angle to the chord. Conversely, the perpendicular to a chord from the centre bisects the chord.

·         Equal chords of a circle are equidistant from the centre. Conversely, the chords that are equidistant from the centre are equal.

·         If two chords say AB and CD of a circle intersect each other internally or externally at point E, then

\[AE\times EB=DE\times EC\]

 

Tangents to a Circle

·         The tangent at any point of a circle is perpendicular to the radius through the point of contact,

i.e.,             \[OT\bot PT.\]

·         If two tangents are drawn to a circle from an outside point, the length of the tangents from the external point to their respective points of contact are equal,

i.e.,             \[PA=PB\]

·         The angle which a chord makes with a tangent at its point of contact is equal to any angle in the alternate segment.

\[\angle PTA=\angle ABT,\] where AT is the chord and PT is the tangent to the circle.

·         If FT is a tangent (with P being an external point and T being the point of contact) and PAB is a secant to circle (with A and B as the points where the secant cuts the circle), then \[P{{T}^{2}}=PA\times PB\]

Pair of Circles

·         If two circles touch each other, the point of contact of the two circles lies on the straight line passing through the centres of the circles, i.e., the points A, C and B are collinear.

·         In a given pair of circles, there are two types of tangents. The direct tangents and the cross (or transverse) tangents. In the figure, AB and CD are the direct tangents and EH and GF are the transverse tangents.

·         When two circles of radii \[{{r}_{1}}\]and \[{{r}_{2}}\]have their centres at a distance d, the length of the direct common tangent

\[=\sqrt{{{d}^{2}}-{{({{r}_{1}}-{{r}_{2}})}^{2}}}\] and the length of transverse tangent

\[=\sqrt{{{d}^{2}}-{{({{r}_{1}}+{{r}_{2}})}^{2}}}.\] If the two circles touch, then \[d={{r}_{1}}+{{r}_{2}}.\]

The sum of pair of opposite angles of a cyclic quadrilateral is equal to 180°.

i.e., \[\angle A+\angle C=\angle B+\angle D=180{}^\circ \]

·         A tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.

 

OTHER PROPERTIES OF CIRCLE

·         The angle subtended by an arc at the centre of the circle is twice the angle substanded by the same arc at any other point on the circle.

·         Angles inscribed in the same segment (or same arc) of a circle are equal

·         Angle in a semi-circle is a right angle.

·         A quadrilateral inscribed in a circle is called a cyclic quadrilateral and its pair of opposite angles is 180° and vise-versa.

·         Exterior angle of acyclic quadrilateral is equal to the interior opposite angle