project on circle

8/11/2019 Project on Circle

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Definition of a circle A circle is a set of those points in a plane that are at a given constantdistance from a givenfixed point in theplane. The fixedpoint is called thecentre of the circle and the constant

distance of every point on the circlefrom its centre is called the radiusof the circle. A circle with centre O andradius r is denoted by C(O, r).

e !now that in a circle," Any line segment with one end pointat the centre of a circle and the other on the circle is called the radius of that circle." Any line segment passing through the centre of the circle having its both ends on thecircle is called the diameter of that circle." Any line segment having its both ends on the circle is !nown as the chord of a circle.

#iameter is the longest chord of the circle.$t is important to note that chord may not pass through the center point of the circle but adiameter always passes through the center, dividing it into two halves.

Perpendiculars From the Center to the Chord1.The perpendicular from the center to the chord of a circle, bisects the chord.To understand this,a. #raw a circle of a !nown radius and mar! the centre as O.b. #raw any chord A% in the circle and measure its length.c. &rom the centre O draw a line perpendicular to the chord meeting at point C as shownin the figure below.d. 'easure the lengths of AC and C%. %oth will measure the same proving that a linedrawn perpendicular from the centre to the chord bisects the chord into two.

2. The line joining the centre to the midpoint of a chord to the centre of the circle isalways perpendicular to the chord.To understand this,a. #raw a circle of any radius and mar!its centre as O.b. #raw a chord of !nownmeasurement on the circle and bisect itat point '.c. oin the centre of the circle to thepoint '.d. 'easure the angle O'A and O'%which is **, proving the line to beperpendicular to the chord.



. Two chords in a circle subtending e!ual angles at the centre aree!ual.

Distances of "!ual chords from the centre of the circle As we now !now that there can be as many chords in the circle and as we ta!e the chordfarther from the centre of the circle, itstarts getting smaller.

+ow, suppose we have two eualchords in a circle, to find out whether the chords are at eual distances fromthe centre point, we will do the followingexercisea. #raw a circle of !nownradius and mar! its centre point as O.b. #raw two chords A% and C# of euallength i.e. A% - C#.c. &rom the centre O drawperpendiculars O' and O+ meeting A% and C# at ' and + respectively.d. 'easure O' and O+, which are eual and prove that two eual chords in a circle lie at

same distance from the centre.Therefore, we conclude that eual chords of a circle are euidistant from the centre

. Chords of a circle which are euidistant from the center are eual.

To understand this,a. #raw a circle of !nown radius with the center point mar!ed as O.b. &rom the center draw two line segments O' and O+ of length such O' - O+ and lessthan the radius of the circle.c. Through ' draw a chord A% such that O' is perpendicular to A%. /imilarly draw chordC#.d. +ow measure A% and C#, which are eual proving that chords euidistant from the

center of the circle are eual

#ome $ngle Properties of the Circle

An angle and a circle in the same planemay or may not intersect.



Central $ngles and their inscribed arcs%Central $ngle%

An angle with itsvertex at thecenter of a circleis called a centralangle of the circle.

&ntercepted $rc %

An arc is intercepted by an angle when (i) the end0points of the arc lie on the

arms of angle, (ii) each arm of the angle contains at least one end0point, and(iii) except for the end0points, the arc lies in the interior of the angle.



&ntercepted $rc by the Central $ngle$ntercepted arcs of eual central angles of a circle are congruent

Degree 'easure of an $rc #egree 'easure of an arc is themeasure of its central arc in degrees.

&nscribed $ngles and their inscribed arcs1 &nscribed $ngle

An angle is inscribed in a circle when(i) its vertex lies on the circumferenceof the circle and (ii) each of its twoarms intersects the circle in twodistinct points



&ntercepted $rc by &nscribed $ngle2i!e in case of central angles, the intercepted arcs of eual inscribed angles of a circle arecongruent.

Conversely, congruent arcs ma!e eual inscribed angles in the remaining circle.

The above also proves that Angles inscribed in the same arc of the circle are eual.

$ngles &nscribed in the same arc of the circle

Two angles are said to be inscribed in the same arc if the intercept the same arc.



(elationship between the central angle and inscribed angle of an arc%

$n a circle, the inscribed angle of an arc is half the measure of its central angle.

The above helps us to prove that the angle inscribed in a semicircle is always *

*

.



A semicircle has a central angle - 34** (as the arc5s end points and the center of the circle

are always on the same line), hence its inscribed angle is always * *.

Cyclic 6uadrilateral and its Angles1 Cyclic )uadrilateral

A uadrilateral with all of its vertices on the circle is said to be a cyclic uadrilateral

The opposite angles of a cyclic uadrilateral aresupplementary.



Tangents to a Circle A line may intersect a circle in the same plane in one or two distinct points. $t may also not

intersect the circle. #ecant A line that intersects a circle in two distinct points is called a secant to the circle. Tangent

A line that intersects a circle in exactly one point is called a tangent to the circle

.

project on circle

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