In the given fig., AB and AC are two equal chords of length 7cm and ∟BAC=60 degrees. Find the area of the shaded region A,C,B in clockwise direction.

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- Feb 25th 2010, 07:49 AMsnigdhaFind the area of the shaded region...
In the given fig., AB and AC are two equal chords of length 7cm and ∟BAC=60 degrees. Find the area of the shaded region A,C,B in clockwise direction.

- Feb 25th 2010, 08:50 AMqmech
The problem would be easy if you knew the radius of the circle.

To find the radius, draw a radius from the center to A and to B. Clearly this is a 30, 30, 120 triangle. Since you know AB=7, the law of cosines can get you the radius.

Once you have r, you have 2 triangles and a section of the circle of degree measure 120 to add up. - Feb 25th 2010, 09:01 AMArchie Meade
Hi snighda,

Since AB and AC are equal, the geometry is symmetrical with respect to

the centre of the circle.

if you join all 3 vertices A, B, C to the circle centre O,

you will have two isosceles triangles ABO and ACO, with opposite angles of 60/2 = 30 degrees and obtuse angle 180-2(30) = 120 degrees

and a segment, with a 60(2) = 120 degree angle at O.

If you use the Sine Rule for one of the isosceles triangles,

then you can find the radius, easily solving for segment and triangle areas.

The sum of all the areas of all three regions is the shaded area.

$\displaystyle \frac{R}{Sin30^o}=\frac{7}{Sin120^o}$

$\displaystyle R=\frac{7Sin30^o}{Sin120^o}$