Question 1110581: What percent of the area for the circumscribed circle is occupied by the triangle? Vertices are (-6,3), (4,13), and (10,-5). The area of the triangle is 120.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Answer #1 -- using the formula for the area of a triangle's circumcircle.
The radius of the circumcircle of a triangle, in terms of the lengths of the three sides a, b, and c, is given by the formula
where A is the area of the triangle.
The Pythagorean Theorem aka distance formula for the given coordinates of the vertices of the triangle gives the three side lengths as sqrt(200), sqrt(320], and sqrt(360). The radius of the circumcircle is then
The area of the triangle is given to be 120. So the ratio of the area of the triangle to the area of the circumcircle is
 = 0.38197 to 5 decimal places.
Answer #2 -- by finding the center of the circumcircle.
The center of the circumcircle is equidistant from the three vertices, so it lies on the perpendicular bisectors of all three sides of the triangle. So we can find the center of the circumcircle by finding the intersection of any two of those three perpendicular bisectors.
Call the three vertices A(-6,3), B(4,13), and C(10,-5).
Side AC has slope -1/2 and midpoint (2,-1); the equation of its perpendicular bisector is  or  .
Side AB has slope 1 and midpoint (-1,8); the equation of its perpendicular bisector is  or  .
Quick basic algebra shows the intersection of those two perpendicular bisectors to be the point (4,3).
Then the distance formula again verifies that the distance from (4,3) to each of the vertices of the triangle is 10.
And then once again we have the answer that the ratio of the area of the triangle to the area of its circumcircle is
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