two chords ab and CD length 5 cm and 11cm respectively of a circle are parallel to each other and an Arc on opposite side of its centres if the distance between a b and c t 6 centimetre find the radius of the circle
Answers
Let r be the radius of given circle and its centre be O. Draw OM
perpendicular AB and ON perpendicular CD
Since, OM perpendicular AB, ON perpendicular CD
and AB||CD
Therefore, points M, O and N are collinear.So, MN = 6cm
Let, OM = x cm.Then,ON = (6 - x)cm.
Join OA and OC. Then OA = OC = r.
As the perpendicular from the centre to a chord of the circle
bisects the chord.
∴ AM = BM = 1/2AB = 1/2 x 5 = 2.5cm.
CN = DN = 1/2CD = 1/2 x 11 = 5.5cm.
In right triangles OAM and OCN, we have
OA^2 = OM^2 + AM^2 and OC^2 = ON^2 + CN^2
r^2 = x^2 + (5/2)^2 ...(i)
r^2 = (6-x)^2 + (11/2)^2 ....(ii)
From (i) and (ii),we have
x^2 + (5/2)^2 = (6-x)^2 + (11/2)^2
x^2 + 25/4 = 36 + x2 - 12x + 121/4
⇒ 4x^2 + 25 = 144 + 4x^2 - 48x + 121
⇒ 48x = 240
⇒ x = 240/48 ⇒ x = 5
Putting the value of x in euation (i), we get
r^2 = 5^2 + (5/2)^2 ⇒ r^2 = 25 + 25/4
⇒ r^2 = 125/4 ⇒ r = 5√5/2 cm
Therefore, radius of the circle = r = (5√5)/2 cm