Question 1207162
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Draw a circle centered at point A.
Plot points B and C that are on the circle, and that are on the same horizontal level. 
Draw segment BC.
Erase a big chunk of the circle so you're only left with minor arc BC.
The arc has endpoints B and C.


Based on the values mentioned in the instructions, we can say that:
Chord BC is 200 meters
Angle BAC = 80 degrees
{{{
drawing(400,400,-5,5,-4,6,

circle(0,0,0.05),circle(0,0,0.07),circle(0,0,0.09),circle(0,0,0.11),circle(0,0,0.13),circle(0,0,0.15),circle(-2,2,0.05),circle(-2,2,0.07),circle(-2,2,0.09),circle(-2,2,0.11),circle(-2,2,0.13),circle(-2,2,0.15),circle(2,2,0.05),circle(2,2,0.07),circle(2,2,0.09),circle(2,2,0.11),circle(2,2,0.13),circle(2,2,0.15),

line(-2,2,2,2),
line(-2,2,0,0),line(0,0,2,2),

arc(0,0,2*sqrt(8),2*sqrt(8),225,315),

locate(0.1,-0.1,"A"),locate(-1.9-0.5,1.9,"B"),locate(2.1,1.9,"C"),

locate(-1.08+1,2.24+0.18,"200"),locate(-0.56+0.3,1.14+0.1,80^o),

locate(-4.8,-3,matrix(1,4,"Diagram","not","to","scale"))

)
}}}


Let D be the midpoint of segment BC.
This means BD = DC and BD is half of BC
Therefore BD = 200/2 = 100 meters.


Draw a segment AD to form triangles ABD and ACD.
They are congruent right triangles. We can use the SSS (side side side) congruence theorem to prove this claim. 
{{{
drawing(400,400,-5,5,-4,6,

circle(0,0,0.05),circle(0,0,0.07),circle(0,0,0.09),circle(0,0,0.11),circle(0,0,0.13),circle(0,0,0.15),circle(-2,2,0.05),circle(-2,2,0.07),circle(-2,2,0.09),circle(-2,2,0.11),circle(-2,2,0.13),circle(-2,2,0.15),circle(2,2,0.05),circle(2,2,0.07),circle(2,2,0.09),circle(2,2,0.11),circle(2,2,0.13),circle(2,2,0.15),circle(0,2,0.05),circle(0,2,0.07),circle(0,2,0.09),circle(0,2,0.11),circle(0,2,0.13),circle(0,2,0.15),

line(0,0,0,2),line(-2,2,2,2),
line(-2,2,0,0),line(0,0,2,2),
line(-0.35,2,-0.35,1.65),line(-0.35,1.65,0,1.65),

arc(0,0,2*sqrt(8),2*sqrt(8),225,315),

locate(0.1,-0.1,"A"),locate(-1.9-0.5,1.9,"B"),locate(2.1,1.9,"C"),locate(0.1,1.9,"D"),

locate(-1.08,2.24+0.18,"100"),locate(-0.56-0.1,1.14+0.1,40^o),

locate(-4.8,-3,matrix(1,4,"Diagram","not","to","scale"))

)
}}}


Because we have congruent triangles, we know that 
angle DAB = angle DAC = 40
These angles are half of angle BAC = 80.


Let's focus entirely on triangle ABD.
We'll use a trig ratio to determine radius AB.
sin(angle) = opposite/hypotenuse
sin(A) = BD/AB
sin(40) = 100/AB
AB = 100/sin(40)
AB = 155.572382686 approximately is the radius
Please make sure that your calculator is set to degrees mode.


Let's determine the circumference of this circle.
C = 2pi*r
C = 2pi*155.572382686
C = 977.4901090956
I used the calculator's stored version of pi to get the most accuracy possible.


This is the approximate distance around the full circle.
However, we don't want a full circle. 
Instead we just want the arc piece.
80/360 = 2/9
We want 2/9 of a full circle.
(2/9)*977.4901090956 = 217.220024243467


Answer: <font color=red>approximately 217.220024 meters</font>
This is the curved distance from B to C.
Round this value however your teacher instructs.
Your answer may vary depending how you rounded intermediate steps and what level of precision you use for pi.
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