2022 AIME II Problems/Problem 7

Problem

A circle with radius is externally tangent to a circle with radius . Find the area of the triangular region bounded by the three common tangent lines of these two circles.

Solution 1

//Created by isabelchen

size(12cm, 12cm);

draw(circle((0,0),24));
draw(circle((30,0),6));
draw((72/5, 96/5) -- (40,0));
draw((72/5, -96/5) -- (40,0));
draw((24, 12) -- (24, -12));
draw((0, 0) -- (40, 0));
draw((72/5, 96/5) -- (0,0));
draw((168/5, 24/5) -- (30,0));
draw((54/5, 72/5) -- (30,0));

dot((72/5, 96/5));
label("$A$",(72/5, 96/5),NE);
dot((168/5, 24/5));
label("$B$",(168/5, 24/5),NE);
dot((24,0));
label("$C$",(24,0),NW);
dot((40, 0));
label("$D$",(40, 0),NE);
dot((24, 12));
label("$E$",(24, 12),NE);
dot((24, -12));
label("$F$",(24, -12),SE);
dot((54/5, 72/5));
label("$G$",(54/5, 72/5),NWW);
dot((0, 0));
label("$O_1$",(0, 0),S);
dot((30, 0));
label("$O_2$",(30, 0),S);

 (Error making remote request. Unknown error_msg)

To be continued......

~isabelchen

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=7NGkVu0kE08

See Also

These problems are copyrighted © by the Mathematical Association of America.