1982 AHSME Problems/Problem 14

Problem 14:

In the adjoining figure, points $B$ and $C$ lie on line segment $AD$ , and $AB, BC$ , and $CD$ are diameters of circle $O, N$ , and $P$ , respectively. Circles $O, N$ , and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$ . If $AG$ intersects circle $N$ at points $E$ and $F$ , then chord $EF$ has length

$[asy] size(250); defaultpen(fontsize(10)); pair A=origin, O=(1,0), B=(2,0), N=(3,0), C=(4,0), P=(5,0), D=(6,0), G=tangent(A,P,1,2), E=intersectionpoints(A--G, Circle(N,1))[0], F=intersectionpoints(A--G, Circle(N,1))[1]; draw(Circle(O,1)^^Circle(N,1)^^Circle(P,1)^^G--A--D, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F^^G^^O^^N^^P); label("$ (Error compiling LaTeX. Unknown error_msg)A $", A, W); label("$ B $", B, SE); label("$ C $", C, NE); label("$ D $", D, dir(0)); label("$ P $", P, S); label("$ N $", N, S); label("$ O $", O, S); label("$ E $", E, dir(120)); label("$ F $", F, NE); label("$ G $", G, dir(100));[/asy]$

Solution:

Since $GP$ is 15, $AP$ is 75, and $\angle{AGP}=90$ , $AG=15\sqrt{24}$ .

Now drop an altitude from $N$ to $AG$ at point $H$ . $AN=45$ , and since $\triangle{AGP}$ is similar to $\triangle{AHN}$ . $NH=9$ . $NE=NF=15$ so by Pythagorean Theorem, $EH=HF=12$ . Thus $EF=\boxed{24}$ . $\boxed{C}$

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1982 AHSME Problems/Problem 14