Difference between revisions of "2022 AIME II Problems/Problem 7"

Revision as of 18:38, 29 April 2022

Problem

A circle with radius $6$ is externally tangent to a circle with radius $24$ . Find the area of the triangular region bounded by the three common tangent lines of these two circles.

Solution 1

$[asy] //Created by isabelchen size(12cm, 12cm); draw(circle((0,0),24)); draw(circle((30,0),6)); draw((72/5, 96/5) -- (40,0)); draw((72/5, -96/5) -- (40,0)); draw((24, 12) -- (24, -12)); draw((0, 0) -- (40, 0)); draw((72/5, 96/5) -- (0,0)); draw((168/5, 24/5) -- (30,0)); draw((54/5, 72/5) -- (30,0)); dot((72/5, 96/5)); label("$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$",(54/5, 72/5),NW); dot((0, 0)); label("$O_1$",(0, 0),S); dot((30, 0)); label("$O_2$",(30, 0),S); [/asy]$

$r_1 = O_1A = 24$ , $r_2 = O_2B = 6$ , $AG = BO_2 = r_2 = 6$ , $O_1G = r_1 - r_2 = 24 - 6 = 18$ , $O_1O_2 = r_1 + r_2 = 30$

$\triangle O_2BD \sim \triangle O_1GO_2$ , $\frac{O_2D}{O_1O_2} = \frac{BO_2}{GO_1}$ , $\frac{O_2D}{30} = \frac{6}{18}$ , $O_2D = 10$

$CD = O_2D + r_1 = 10 + 6 = 16$ ,

$EF = 2EC = EA + EB = AB = GO_2 = \sqrt{(O_1O_2)^2-(O_1G)^2} = \sqrt{30^2-18^2} = 24$

$DEF = \frac12 \cdot EF \cdot CD = \frac12 \cdot 24 \cdot 16 = \boxed{\textbf{192}}$

~isabelchen

Solution 2

Let the center of the circle with radius $6$ be labeled $A$ and the center of the circle with radius $24$ be labeled $B$ . Drop perpendiculars on the same side of line $AB$ from $A$ and $B$ to each of the tangents at points $C$ and $D$ , respectively. Then, let line $AB$ intersect the two diagonal tangents at point $P$ . Since $\triangle{APC} \sim \triangle{BPD}$ , we have $\frac{AP}{AP+30}=\frac14 \implies AP=10.$ Next, throw everything on a coordinate plane with $A=(0, 0)$ and $B = (30, 0)$ . Then, $P = (-10, 0)$ , and if $C = (x, y)$ , we have $(x+10)^2+y^2=64,$ $x^2+y^2=36.$ Combining these and solving, we get $(x, y)=\left(-\frac{18}5, \frac{24}5\right)$ . Notice now that $P$ , $C$ , and the intersections of the lines $x=6$ (the vertical tangent) with the tangent containing these points are collinear, and thus every slope between a pair of points will have the same slope, which in this case is $\frac{-\frac{18}5+10}{\frac{24}5}=\frac34$ . Thus, the other two vertices of the desired triangle are $(6, 12)$ and $(6, -12)$ . By the Shoelace Formula, the area of a triangle with coordinates $(-10, 0)$ , $(6, 12)$ , and $(6, -12)$ is $\frac12|-120-0-72-72+0-120|=\boxed{\textbf{192}}.$

~A1001

Solution 3

(Taking diagram names from Solution 1. Also say the line that passes through O_1 and is parallel to line EF, call the points of intersection of that line and the circumference of circle O_1 points X and Y.)

First notice that DO_1 is a straight line because DXY is an isosceles triangle(or you can realize it by symmetry). That means, because DO_1 is a straight line, so angle BDO_2 = angle ADO_1, triangle ADO_1 is similar to triangle BDO_2. Also name DO_2 = x. By our similar triangles, BO_2/AO_1 = 1/4 = x/(x+30). Solving we get x = 10 = DO_2. Pythagorean Theorem on triangle DBO_2 shows BD = sqrt(10^2 - 6^2) = 8. By similar triangles, DA = 4*8 = 32 which means AB = DA - DB = 32 - 8 = 24. Because BE = CE = AE, AB = 2*BE = 24. BE = 12, which means CE = 12. CD = DO_2(its value found earlier in this solution) + CO_2(O_2 's radius) = 10 + 6 = 16. The area of DEF is 1/2 * CD * EF = CD * CE (because CE is 1/2 of EF) = 16 * 12 = 192.

~Professor Rat's solution, added by heheman

Video Solution (Mathematical Dexterity)

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

2022 AIME II (Problems • Answer Key • Resources)
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