Circle: 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.
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.
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.
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.
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.
Theorem 9: Angles in the same segment of a circle are equal.
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)
Theorem 11: The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
Theorem 12: If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.
Question 1: Fill in the blanks:
- The centre of a circle lies in …………..of the circle. (exterior/ interior)
- A point, whose distance from the centre of a circle is greater than its radius lies in …………of the circle. (exterior/ interior)
- The longest chord of a circle is a ………..of the circle.
- An arc is a ………….when its ends are the ends of a diameter.
- Segment of a circle is the region between an arc and ……………of the circle.
- A circle divides the plane, on which it lies, in …………….parts.
Answer: (i) interior, (ii) exterior, (iii) diameter, (iv) semicircle, (v) chord, (vi) three
Question 2: Write True or False
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
Answer: True; because the distance between centre and any point on circumference is called radius.
(ii) A circle has only finite number of equal chords.
Answer: False, because a circle can haev infinite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
Answer: False, because a major arc is always bigger than a minor arc, and they cannot be equal.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
Answer: True because diameter is the longest chord of a circle and is always twice the radius.
(v) Sector is the region between the chord and its corresponding arc.
Answer: False, the region between the chord and its corresponding arc is called segment.
(vi) A circle is a plane figure.