Topic: Geometry - Circles
shaalaa.com | Circles (Chord Properties Part 1)
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Circles (Chord Properties Part 1)
Circles (Chord Properties Part 2)
Circles (Chord Properties Part 3)
Circles (Chord Properties Part 4)
Circles (Arc Properties Part 1)
Circles (Arc Properties Part 2)
Circles (Arc Properties Part 3)
Circles (Cyclic Properties Part 1)
Circles (Tangent Properties Part 1)
Circles (Tangent Properties Part 2)
(a) Chord Properties:
- A straight line drawn from the center of a circle to bisect a chord which is not a diameter is at right angles to the chord.
- The perpendicular to a chord from the center bisects the chord (without proof).
- Equal chords are equidistant from the center.
- Chords equidistant from the center are equal (without proof).
- There is one and only one circle that passes through three given points not in a straight line.
(b) Arc and chord properties:-
- The angle that an arc of a circle subtends at the center is double that which it subtends at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal (without proof).
- Angle in a semi-circle is a right angle.
- If two arcs subtend equal angles at the center, they are equal, and its converse.
- If two chords are equal, they cut off equal arcs, and its converse (without proof).
- If two chords intersect internally or externally then the product of the lengths of the segments are equal.
(c) Cyclic Properties:
- Opposite angles of a cyclic quadrilateral are supplementary.
- The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle (without proof).
(d) Tangent Properties:
- The tangent at any point of a circle and the radius through the point are perpendicular to each other.
- If two circles touch, the point of contact lies on the straight line joining their centers.
- From any point outside a circle two tangents can be drawn and they are equal in length.
- If a chord and a tangent intersect externally, then the product of the lengths of segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
- If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.
Note: Proofs of the theorems given above are to be taught unless specified otherwise.