SOLUTION: The circumference and radius of a circle "T" are unknown. Two point A and B lie on circle whose arc length is 200 feet but the length of AB chord is 150 feet. (A) Find the area of

In a coordinate plane make a sketch: draw a circle T (with unknown radius r);

mark the appropriate points A and B  (A in Q1, B in the Q4 symmetrically to A about the x-axis).

Draw the chord AB.


Let the central angle for the arc AB be  radians.

So from the given part, you have these two equations

     = 200 feet,  or    = 100         (1)  (from the arc length)

and

     = 150,  or   = 75.      (2)  (from the chord length).


Dividing equation (2) by equation (1), you get

     = ,  or   = .    (3)


This equation is not algebraic equation; it is transcendent equation and can not be solved 
by the Algebra methods.  But it can be solved numerically with any prescribed precision.


There are many solvers for solving such equations in the Internet.

I used one such free of charge common use solver at this link

https://www.wolframalpha.com/widgets/view.jsp?id=a7d8ae4569120b5bec12e7b6e9648b86


It gave me the value of  = 1.2757  radians.


        To check it,  compare both sides of equation (3)

            sin(1.2757) = 0.9568;  0.75*1.2757 = 0.9568,  

        so these values are consistent.



Next, knowing , we can find "r" from equation (1)

    r =  =  = 78.3883  (rounded).



    +--------------------------------------------------------------+
    |   Now we have everything to answer the problem's questions.  |
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Question (A)


    The area of the segment is the area of the sector minus the area of the triangle

        area of the segment =  -  = 

                            =  -  = 

                            = 7838.8264 - 1709.7699 = 6129.0565 square units  (rounded)    ANSWER



Question (B)


    After solving the problem, I just know the radius of the circle; therefore, I can calculate
    the area of the circle.  It is

        area of the circle =  =  = 19304.2084  (rounded).    ANSWER