Question 1180131
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Find the diameter of the circle that can be circumscribed around a triangle that has two 13-inch sides
and one 10-inch side.
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There is a remarcable formula, connecting the area of a triangle, its sides and the radius of the circumscribed circle.


This formula is  S = {{{(abc)/(4R)}}}      (1)


where "a", "b", "c" are the triangle' side lengths, S is the triangle formulas and R is the radius of the circumscribed circle.


For the proof of this formula see the lesson

    - <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-radius-of-the-circumscribed-circle.lesson>Proof of the formula for the radius of the circumscribed circle</A>

in this site.


The formula works and is valid for ANY TRIANGLE.


By knowing the side lengths of the given triangle, we can calculate its area using the Hero's formula.

But in the given case, we can calculate the area of the triangle by even simpler way as half the product of the base length (10 inches)
and the height drawn to the base.  The height is, obviously, 12 = {{{sqrt(13^2-5^2)}}}, therefore, the area of the triangle is

    S = {{{(1/2)*10*12}}} = 60 sq. inches.


Now from the formula (1), the radius of the circumscribed circle is 

    R = {{{(abc)/(4S)}}} = {{{(13*13*10)/(4*60)}}} = {{{169/24}}}.


Therefore, the diameter of the circumscribed circle is  d = 2R = {{{169/12}}}.    <U>ANSWER</U>
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Solved.