1.In the figure, OD is perpendicular to chord AB of a circle whose centre is O. If BC is diameter; prove that CA = 2OD.
2.   l is a line intersecting two concentric circles having common centre O, at A, B, C and D. Prove that AB = CD.
3.   AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that EB = ED.
4.   If O be the centre of the circle, find the value of �x� in each of the following figures.
5.   Prove that equal chords of a circle subtend equal angles at the centre.
6.   The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. Prove it.
7.   Prove that equal chords of a circle (or congruent circles) are equidistant from the centre (or centres).
8.   In the figure, OD is perpendicular to the chord AB of a circle with centre O. If BC is a diameter, show that AC || OD and AC = 20D.
           Hint: ∴ OD ⊥ AB               therefore; D is the mid-point of AB.
9.   If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.
10.   Show that the angles in the same segment of a circle are equal.

Asked by aahnik.mohanty | 30th Dec, 2017, 10:21: AM

Expert Answer:

begin mathsize 16px style Since space OD perpendicular AB space and space the space perpendicular space drawn space from space the space centre space to space straight a space chord
bisects space the space chord comma
straight D space is space the space midpoint space of space chord space AB.
Also comma space since space straight O space is space the space centre space of space the space circle comma space it space is space also space the space midpoint space of space diameter space BC.
Thus comma space in space triangle space ABC comma space
straight D space and space straight O space are space midpoints space of space AB space and space BC space respectively.
rightwards double arrow OD space is space parallel space to space AC
And comma space OD equals 1 half AC
rightwards double arrow AC equals 2 OD end style
 
 
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Answered by Rashmi Khot | 30th Dec, 2017, 03:48: PM