Two semicircles are constructed as shown in the figure. The chord PQ of the greater circle touching the smaller circle and is
parallel to the diameter of larger circle. If the length of PQ is 10 units, then the area between the semicircles is 5k 7. Find the
value of 'k' ('O' is the centre of big circle).
Answers
Given : Two semicircles are constructed as shown in the figure.
The chord PQ of the greater circle touching the smaller circle and is
parallel to the diameter of larger circle
length of PQ is 10 units,
area between the semicircles is 5k/7
To Find : Value of k
Solution:
Let say Radius of Bigger Semi Circle = 2R
Then Diameter of Smaller Semi Circle = 2R
Hence Radius of smaller Semi circle = R
PQ is chord hence perpendicular bisector of chord PQ meet at O
Let say M is mid point of PQ
=> OM = R as PQ is tangent to smaller semicircle
PM = QM = PQ/2 = 10/2 = 5
PQ = 2R
PQ² = PM² + OM²
=> (2R)² = 5² + R²
=> 3R² =5²
=> R = 5/√3
=> 2R = 10/√3
Area of Bigger Semicircle = (1/2)π(10/√3)² = 50π/3
Area of Smaller Semicircle = (1/2)π(5/√3)² = 25π/6
area between the semicircles = 50π/3 - 25π/6
= 75π/6
= 25π/2
using π = 22/7
= 275/7
= 5*55/7
Compare with 5k/7
=> k = 55
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