document.write( "Question 1105670: How do you find the radius of a circle inscribed in a scalene triangle \n" ); document.write( "

Algebra.Com's Answer #720534 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here is a scalene triangle (ABC) with an inscribed circle centered at O.
\n" ); document.write( "

.
\n" ); document.write( "Here is the same sketch with the line segments connecting the circle center
\n" ); document.write( "with the vertices of the triangle,
\n" ); document.write( "and with the points of tangency with the triangle sides.
\n" ); document.write( "

.
\n" ); document.write( " $\"green%28OD%29\"$ , $\"green%28OE%29\"$ , and $\"green%28OF%29\"$ ,
\n" ); document.write( "connect circle center O to the points of tangency.
\n" ); document.write( "They are radii of the circle, $\"R=OD=OE=OF\"$
\n" ); document.write( "and they are perpendicular to the sides of the triangle.
\n" ); document.write( "So, they are the heights of triangles ABO, BCO, and ACO.
\n" ); document.write( "
\n" ); document.write( "If you know $\"AB=c\"$ , $\"BC=a\"$ , and $\"AC=b\"$ ,
\n" ); document.write( "the lengths of the sides of the triangle,
\n" ); document.write( "you can calculate the area of triangle ABC
\n" ); document.write( "as the sum of the areas of triangles ABO, BCO, and ACO:
\n" ); document.write( " $\"%28AB%2BBC%2BAC%29OD%2F2\"$ or $\"%28a%2Bb%2Bc%29R%2F2\"$ .
\n" ); document.write( "You could also calculate the area of the triangle using Heron's formula as
\n" ); document.write( " $\"sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29\"$ , where $\"s=%28a%2Bb%2Bc%29%2F2\"$ is the semi-perimeter.
\n" ); document.write( "
\n" ); document.write( "If you do not know the semi-perimeter (or the perimeter, or the lengths of all 3 sides of the triangle),
\n" ); document.write( "as long as you have enough information to determine what triangle is congruent to that one,
\n" ); document.write( "there is a way to find those lengths,
\n" ); document.write( "and you could use a similar strategy after you find the needed length(s).
\n" ); document.write( "
\n" ); document.write( "The same strategy would work for any triangle, scalene or otherwise.
\n" ); document.write( "
\n" ); document.write( "If you do not have enough information to find the needed lengths,
\n" ); document.write( "you may know that the triangle is scalene,
\n" ); document.write( "but you do not know the triangle, and cannot find the radius.
\n" ); document.write( "For example, knowing only that the side lengths are in the ratio 3:4:5, or that all 3 angles have different measures would let you know the triangle is scalene, but there would be no way to calculate the radius of the inscribed circle. \n" ); document.write( "

\n" );