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The triangle  shown is a right triangle. The semicircles have the sides of the triangle as diameters. The areas of two of the semicircles are shown. What is the area of the third semicircle? 

 

[asy]
unitsize(0.2 inch);

pair A=4*dir(150),B = 4*dir(-150),C = 4*dir(-30),
P = (A+B)/2 + (-2,0),Q = (B+C)/2 + (0,-2*sqrt(3));
fill(arc((A+B)/2,2,90,270)--cycle,black); fill(arc((C+B)/2,2*sqrt(3),180,360)--cycle,black);
fill(arc((A+C)/2,4,-30,150)--cycle,black);
path D = (-1/2,-1/2)--(-1/2,1/2)--(1/2,1/2)--(1/2,-1/2)--cycle;
fill(shift((-4,0))*D,white); fill(shift((0,-4))*D,white);
label("7",(-4,0)); label("13",(0,-4));
label("$A$",A,NW); label("$B$",C,SE); label("$C$",B,SW); label("$P$",P,W); label("$Q$",Q,S);
[/asy]

 Oct 31, 2020
 #1
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+1

area = 20 units squared

 

 Oct 31, 2020

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