Question

Three statements are given below:
I. If a diameter of a circle bisects each of the two chords of a circle, then the chords are parallel.
II. Two circles of radii 10 cm and 17 cm intersect each other and the length of the common chord is 16 cm. Then, the distance between their centres is 23 cm.
III. L is the line intersecting two concentric circles with centre O at points A, B, C and D as shown Then, AC = DB.
Which is true?
(a) I and II
(b) I and III
(c) II and III
(d) II only

Answer 1

Option (b) is correct.
Parts (I) and (III) are clearly true.
Let us examine (II).
Let B and C be the centres of two circles of radii 10 cm and 17 cm, respectively. Let AD be the common chord cutting BC at E.
Then AE = ED = 8 cm

Thus, we have:
$\mathrm{BE}=\sqrt{{\left(10\right)}^2-{\left(8\right)}^2}=\sqrt{100-64}=\sqrt{36}=6\mathrm{cm}$
$\mathrm{CE}=\sqrt{{\left(17\right)}^2-{\left(8\right)}^2}=\sqrt{289-64}=\sqrt{225}=15\mathrm{cm}$
∴ BC = BE + CE = (6 + 15) cm = 21 cm.
But it is given that BC is 23 cm.
So, the given statement is wrong.