Question 1112853: A quadrilateral circumscribing a circle has 3 sides; 5 inches, 4 inches and 5.74 inches. Find the fourth side if the radius of the circle is 2.5 inches. Find also the area of the quadrilateral. Please explain.
Found 2 solutions by rothauserc, KMST:
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! Use the Pitot Theorem
:
5 + 5.74 = 4 + x
:
10.74 = 4 + x
:
x = 10.74 - 4 = 6.74 inches
:
A = r * s, where r is the radius and s is the semi-perimeter
:
A = 2.5 * (5 +5.74 +4 +6.74) / 2 = 26.85 square inches
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Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! For the quadrilateral in the sketch below,  .
The Pitot theorem says that for a circumscribed quadrilateral with side lengths a, b, c and d in that order,
 .
(In words, the sum of lengths of opposite sides is the semiperimeter).
Then, the area is  where  =radius of incircle,
but  .
The question did not state that the side length were given in order going around the quadrilateral,
but if we assume that another pair of sides are opposite,
the inequality above is not true,
meaning that a circumscribed quadrilateral side lengths could be
5, 4, 5.74, and 3.26, or
5, 4, 5.74, and 4.74, but then the circle would have to be smaller.
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