write all the theorem of circles of class 9th
Answers
write all the theorem of circles
Theorem 1: Equal chords of a circle subtend equal angles at the centre.
Theorem 2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
PERPENDICULAR FROM THE CENTRE TO A
CHORD
Theorem 3 : The perpendicular from the centre of a circle to a chord bisects the chord.
Theorem 4 : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Theorem 5 : There is one and only one circle passing through three given non-collinear points.
Remark : If ABC is a triangle, then by above given Theorem there is a unique circle passing through the three vertices A, B and C of the triangle. This circle is called the circumcircle of the AABC. Its centre and radius are called respectively the circumcentre and the circumradius of the triangle.
EQUAL CHORDS AND THEIR DISTANCES FROM THE CENTRE
Theorem 6 : Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
Theorem 7 : Chords equidistant from the centre of a circle are equal in length.
ANGLE SUBTENDED BY AN ARC OF A CIRCLE
Result : Congruent arcs (or equal arcs) of a circle subtend equal angles at the centre.
Theorem 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.
Note : Theorem gives the relationship between the angles subtended by an are at the centre and at a point on the circle.
ANGLE FORMED IN THE SEGMENT
Theorem 9: Angles in the same segment of a circle are equal.
Note : Angle in a semicircle is a right angle.
Theorem 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).
CYCLIC QUADRILATERAL:
A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle.
Maths Class 9 Notes - Circles
Theorem 11: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
Theorem 12 : If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.
Answer:
1. The angle at the centre is twice the angle at the circumference
2. The angle in a semicircle is a right angle
3. Angles in the same segment are equal
4. Opposite angles in a cyclic quadrilateral sum to 180°
5. The angle between the chord and the tangent is equal to the angle in the alternate segment
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