Fill in the blanks:
(i) The center of a circle lies in _ of the circle. (exterior/ interior)
(ii) A point, whose distance from the center 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) A 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.
Write True or False: Give reasons for your Solutions
(i) Line segment joining the center 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 a diameter of the circle.
(v) The sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
If a line intersects two concentric circles (circles with the same center) with center O at A, B, C, and D, prove that AB = CD (see Fig. 10.25).
Three girls Reshma, Salma, and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.
Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D, and P, Q respectively (see Fig. 10.40). Prove that ∠ACP =∠QCD.
Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its center. If the distance between AB and CD is 6, find the radius of the circle.
Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the center.
AC and BD are chords of a circle that bisects each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.
Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lies on the two circles. Prove that BP = BQ.
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