# Find the area of the shaded region...

• Feb 25th 2010, 07:49 AM
snigdha
Find 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 AM
qmech
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 AM
Quote:

Originally Posted by snigdha
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.

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.

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

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