Question

Express answer in exact form.

A segment of a circle has a 120 arc and a chord of 8sqrt3 in. Find the area of the segment.

help me someone !!!

Answer 1

Drawing a diagram you end up ewith a 8sqrt3 chord across a circle with two radii connecting to it. We don't know the radii. We do know the angle of 120 degrees. If We draw a line to the midpoint of the chord we have 2 90 degree angles and two 4sqrt3 parts to the split 8sqrt3 chord.

We are splitting the central angle in two and have 2 60 degree angles. This gives us 30 as the left over angle. We can use tangent to figure out the missing side to get the area.

\[tan(30) = x / (4 \sqrt {3})\]
\[ 4 \sqrt {3} tan(30) =x\]
\[x = 4\]

Now we can figure out the area. 2 similar triangles make a square. 2(1/2)base*height.
\[4*4 \sqrt {3} = 16 \sqrt {3}\]

Answer 2

I'm assuming we're looking for the enclosed area between the center and chord along the 120 angle. It's been a while since geometry :)

Answer 3

That's in inches btw ;)