SOLUTION: A circle passes through A and B, and is tangent to line CD. Find its area, in cm^2. Diagram: https://imgur.com/a/XwWg5Kc

Algebra ->  Surface-area -> SOLUTION: A circle passes through A and B, and is tangent to line CD. Find its area, in cm^2. Diagram: https://imgur.com/a/XwWg5Kc      Log On


   

Question 1205840: A circle passes through A and B, and is tangent to line CD. Find its area, in cm^2.
Diagram: https://imgur.com/a/XwWg5Kc
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
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A circle passes through A and B, and is tangent to line CD. Find its area, in cm^2.
Diagram: https://imgur.com/a/XwWg5Kc
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        The problem does not tell it,  by it is clear from the plot,
        and  I  will assume it,  that  AB  is parallel to  CD. 


Make a sketch for better understanding.  
Let "r" be the radius of the circle, in centimeters.


The center of the circle lies on the perpendicular bisector to AB,
and the distance from the center to AB is  sqrt%28r%5E2-2%5E2%29 = sqrt%28r%5E2-4%29 cm.


Therefore, this equlation is hold 

    r + sqrt%28r%5E2-4%29 = 10 cm


Our task is to solve it to determine the rasdius r.
It is easy. Move r from left side to right

    sqrt%28r%5E2-4%29 = 10-r.


Square both sides

    r%5E2+-+4+ = %2810-r%29%5E2

    r%5E2+-+4 = 100+-+20r+%2B+r%5E2


 Cancel r%5E2 in both sides and simplify

    20r = 100 + 4

    20r = 104

      r = 104%2F20 = 26%2F5 cm = 5.2 cm.


Now the area of the circle is  

    pi%2Ar%5E2 = 3.14159265%5E2%2A5.2%5E2 = 266.8741 cm^2 (rounded).    ANSWER

Solved.