Question 1201843: In the figure below, ADC is a chord of a circle centre O and passing
through the points A, B and C. BD is a perpendicular bisector
of the chord AC. AD=8 cm and BD=2 cm. Calculate the area of the
minor segment ABCD.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52786)   (Show Source):
You can put this solution on YOUR website! .
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Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
In the figure below, ADC is a chord of a circle centre O and passing
through the points A, B and C. BD is a perpendicular bisector
of the chord AC. AD=8 cm and BD=2 cm. Calculate the area of the
minor segment ABCD.
I'll just do the first one for you.
Here are the steps to find the area of a segment of a circle.
1. Identify the radius of the circle and label it 'r'.
2. Identify the central angle AOC made by the arc of the segment and label it
 .
3. Find the area of triangle AOC using the formula  or  .
4. Find the area of the sector OABC using the formula
 , if θ is in degrees (or)
 , if θ is in radians.
5. Subtract the area of the triangle OAC from the area of the sector OABC to
find the area of the segment ABCD.
So we need to find radius r and angle θ.
Draw in OD (in green). Since OA, OB, OC are all radii, with length r,
and since BD=2 cm, OD = OB-BD = r-2
To find radius r, we use the Pythagorean theorem on right triangle OAD:
To find θ, we use:
Now we go back to the given figure:
For step 3, we find the area of triangle OAC either by
or  .
For step 4, we find the area of the entire sector ABCO either by
, if θ is in degrees
^""/360^o)*pi*r^2}}}  
(or)
, if θ is in radians.
 
Notice there's a slight difference between those two values.
Finally we do step 6 and subtract the area of the triangle from the area of the
sector
(or)
Edwin
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