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Arcs AB and CD have a common center O and have the lengths shown. Find the area of circular sector OAB. 
Arc AB=4. AC=2, arc CD=7
 

 Feb 17, 2024
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Let r=OA

Since central angles AOB and COD are equal, the length of each arc is proportional to the corresponding radius. Thus, 4/r=7/(r+2)
Cross-multiplying, we get 4r+8=7r so 3r=8 and r=8/3 Then the area of the circle with radius r is pir^2=64/9pi

The proportion of AOB to 360 is equal to the proportion of arc AB to the circumference of the whole circle, which is

4/(2rpi)=4(2pi*8/3)=3/(4pi)
so the area of circular sector OAB is 3/(4pi)*64/9 (pi)=16/3
 

 Feb 18, 2024

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