plzz solve this ques
Areas related to Circles chapter
Answers
= (22/7)*(25^2 - 20.5^2)
= 643.5
FINDING THE RADII
Let O be the centre of the small circle .
Let AO be r.
Then , diameter AD = 2 r
Now join O and E .
OE = AO = r [ radius ]
In triangle OCE ,
OE = radius
Let the radius of the big circle be R
EC = big circle's radius - Crescent's length
= R - 5 [ given : 5 ]
OC = big circle's radius - small circle's radius
= R - r
Now , ∠ECO = 90°
By Pythagoras theorem :
EC² + OC² = OE²
==> ( R - 5 )² + ( R - r )² = r²
==> R² + 5² - 10 R + R² + r² - 2 Rr = r²
Cancelling r² both sides we get :
==> 2 R² + 25 - 10 R - 2 Rr = 0 .............................(1)
Note that AB - AO = 9 [ given ]
==> 2 R - 2 r = 9
==> 2 r = 2 R - 9
==> r = R - 9/2 ...............................(2)
Insert this value in (1) :
2 R² + 25 - 10 R - 2 R r = 0
==> 2 R² + 25 - 10 R - 2 R ( R - 9/2 ) = 0
==> 2 R² + 25 - 10 R - 2 R² + 9 R = 0
Cancel 2 R² :
==> 25 - 10 R + 9 R = 0
==> - R = - 25
==> R = 25
We got : r = R - 9/2
==> r = 25 - 9/2
==> r = ( 50 - 9 ) / 2
==> r = 41 / 2
==> r = 20.5
AREA OF SHADED REGION
Area of shaded region = area of the big circle - small circle
==> π R² - π r²
==> π ( R² - r² )
==> π ( 25² - ( 20.5 )² )
==> π ( 625 - 420.25 )
==> 22/7 × 204.75
==> 4504.5/7
==> 643.5
ANSWERS
The radius of small circle is 20.5 cm
The radius of big circle is 25 cm
The area of crescent is 643.5 cm²
Hope it helps you:-)
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