document.write( "Question 1143730: Determine the equation of the circle inscribed in a triangle, if the triangle has its sides on the lines y=0, 4x+3y-50=0, and 3x-4y=0. Sketch the graph. \n" ); document.write( "
Algebra.Com's Answer #764643 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!
Here is a VERY DIFFERENT solution method, no better and no worse than the one shown by tutor @ikleyn. It shows nicely how two completely different methods can be used to solve a problem, if you have the tools you need.

\n" ); document.write( "It should be noted that this method is possible for this example only because the triangle is a right triangle with one side on the x-axis. There are no such restrictions on the general method shown by @ikleyn.

\n" ); document.write( "The equations of the lines containing the sides of the triangle are
\n" ); document.write( "(1) \"y+=+0\"
\n" ); document.write( "(2) \"y+=+%28-4%2F3%29x%2B50%2F3\"
\n" ); document.write( "(3) \"y+=+%283%2F4%29x\"

\n" ); document.write( "The slopes of (2) and (3) show that the triangle is a right triangle; and the slope of (3) tells us that the right triangle has side lengths in the ratio 3:4:5.

\n" ); document.write( "The x-intercept of line (2) is (12.5,0). That means the hypotenuse of the triangle is 12.5. And that, along with the known 3:4:5 ratio of the side lengths, tells us the lengths of the two legs are 7.5 and 10.

\n" ); document.write( "So the perimeter of the triangle is 7.5+10+12.5 = 30; and the area is (1/2)(7.5)(10) = 75/2.

\n" ); document.write( "But the area is also one-half the perimeter times the radius of the inscribed circle, so we can find the radius of the circle:
\n" ); document.write( "\"%281%2F2%29%2830%29r+=+75%2F2\"
\n" ); document.write( "\"15r+=+75%2F2\"
\n" ); document.write( "\"r+=+5%2F2\"

\n" ); document.write( "So we have the radius of the circle; now we need to find the center, (h,k).

\n" ); document.write( "Since the radius is 5/2 and one side of the triangle is on the line y=0, we know the y coordinate of the center is 5/2. So the center of the circle is (h,5/2).

\n" ); document.write( "To find the x coordinate of the center, we can use the fact that the measure of the angle between the x-axis and the line from the origin to the center of the circle is half the measure of the angle of the triangle at the origin.

\n" ); document.write( "\"tan%28theta%2F2%29+=+sqrt%28%281-cos%28theta%29%29%2F%281%2Bcos%28theta%29%29%29\"
\n" ); document.write( "

\n" ); document.write( "So

\n" ); document.write( "\"k%2Fh+=+%285%2F2%29%2Fh+=+1%2F3\"
\n" ); document.write( "\"h+=+15%2F2\"

\n" ); document.write( "We now have the radius of the circle and the coordinates of the center of the circle, so we can write the equation of the circle:

\n" ); document.write( "center(15/2,5/2), radius 5

\n" ); document.write( "\"%28x-15%2F2%29%5E2%2B%28y-5%2F2%29%5E2+=+%285%2F2%29%5E2\" \n" ); document.write( "
\n" );