In the given figure O Is the centre of the two concentric circles. A line 7' cuts the
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circles at A, B, C and D as shown in the figure. OP is perpendicular to AD
Given OA = 34 cm, OP = 16 cm and AB = 18 cm
Find:
(1) length of chord AD
(ii) length of chord BC
(iii) radius of the smaller circle
Answers
Answer:
Step-by-step explanation:
(i) length of chord AD is 60 cm
(ii) length of chord BC is 24 cm
(iii) radius of the smaller circle OC is 20 cm
Step-by-step explanation:
given : OA = 34 cm
OP = 6 cm and AB = 18 cm
let the BP = x
in triangle OAP
using the pythagoras theorem
OA² = OP² +AP²
34² = 16² + (18+x)²
1156-256 = 324 +x² +36x
900 - 324 = x² +36x
x² +36x -576 =0
x² + 48x -12x-576 =0
x(x+48) -12 (x+48) =0
(x-12) (x+48)
x= 12 , x= -48
length can never be negative therefore x = 12
BP = PC = 12 cm
now \
length of chord AD
we know that the P is the mid point of the chord AD
therefore , 2AP = AD
2(AB +BP ) = AD
2(18+12) = AD
2(30) = AD
AD = 60 cm
(ii) length of chord BC
similarly 2BP = BC
BP = 12 cm
BC = 12×2 = 24 cm
(iii) radius of the smaller circle
in triangle OPC
using pythagoras theorem
OC² = OP² +PC²
OC² = 16² + 12²
OC² = 256 +144 = 400
OC² = 400
OC = 20 cm
hence , (i) length of chord AD is 60 cm
(ii) length of chord BC is 24 cm
(iii) radius of the smaller circle OC is 20 cm
#Learn more:
In the given figure If a line intersects two concentric circle with Centre O at a, b, c and d prove that AB = CD
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