If the angle subtended by two chords of a circle at the centre are equal the cord are equal

Описание: If the angle subtended by two chords of a circle at the centre are equal the cord are equal

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10.1 Introduction
10.2 ch 10 and Its Related Terms: A Review: The collection of all the points in a plane,
which are at a fixed distance from a fixed point in
the plane, is called a circle.

10.3 Angle Subtended by a Chord at a Point
10.4 Perpendicular from the Centre to a Chord
10.5 Circle through Three Points
The length of the perpendicular from a point to a line is the distance of the
line from the point.
10.7Angle Subtended by an Arc of a Circle: If two chords of a circle are equal, then their corresponding arcs are congruent
and conversely, if two arcs are congruent, then their corresponding chords are
equal.
Theorem 10.8 : The angle subtended by an arc at the centre is double the angle
subtended by it at any point on the remaining part of the circle.

Theorem 10.9 : Angles in the same segment of a circle are equal.
Again let us discuss the case (ii) of Theorem 10.8 separately. H
Theorem 10.10 : If a line segment joining two points subtends equal angles at
two other points lying on the same side of the line containing the line segment,
the four points lie on a circle (i.e. they are concyclic).

Theorem 10.11 : The sum of either pair of opposite angles of a cyclic
quadrilateral is 180º
Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is
180º, the quadrilateral is cyclic



Theorem 10.1 : Equal chords of a circle subtend equal angles at the centre
Theorem 10.2 : If the angles subtended by the chords of a circle at the centre
are equal, then the chords are equal.
Theorem 10.3 : The perpendicular from the centre of a circle to a chord bisects
the chord
Theorem 10.4 : The line drawn through the centre of a circle to bisect a chord is
perpendicular to the chord
Theorem 10.5 : There is one and only one circle passing through three given
non-collinear points.
Theorem 10.6 : Equal chords of a circle (or of congruent ch 10) are equidistant
from the centre (or centres).

Theorem 10.7 : Chords equidistant from the centre of a circle are equal in
length.



Summary
In this chapter, you have studied the following points:
1. A circle is the collection of all points in a plane, which are equidistant from a fixed point in
the plane.
2. Equal chords of a circle (or of congruent ch 10) subtend equal angles at the centre.
3. If the angles subtended by two chords of a circle (or of congruent ch 10) at the centre
(corresponding centres) are equal, the chords are equal.
4. The perpendicular from the centre of a circle to a chord bisects the chord.
5. The line drawn through the centre of a circle to bisect a chord is perpendicular to the
chord.
6. There is one and only one circle passing through three non-collinear points.
7. Equal chords of a circle (or of congruent ch 10) are equidistant from the centre (or
corresponding centres).
8. Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent) are equal.
9. If two arcs of a circle are congruent, then their corresponding chords are equal and
conversely if two chords of a circle are equal, then their corresponding arcs (minor, major)
are congruent.
10. Congruent arcs of a circle subtend equal angles at the centre.
11. The angle subtended by an arc at the centre is double the angle subtended by it at any
point on the remaining part of the circle.
12. Angles in the same segment of a circle are equal.

EXERCISE 10.1
1. Fill in the blanks:
(i) The centre of a circle lies in of the circle. (exterior/ interior)
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in
of the circle. (exterior/ interior)
(iii) The longest chord of a circle is a of the circle.
(iv) An arc is a when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and of the circle.
(vi) A circle divides the plane, on which it lies, in parts.
2. Write True or False: Give reasons for your answers.
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
(ii) A circle has only a finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
(iv) A chord of a circle, which is twice as long as its radius, is the diameter of the circle.
(v) Sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.