Difference between revisions of "User:Azjps/sandbox"

m (hmm)
m (asy testing)
Line 1: Line 1:
 
<center><asy>
 
<center><asy>
size(200);
+
defaultpen(fontsize(11pt));
defaultpen(0.8);
+
size(250);
pair D=(0,0), C=(0,24*3^0.5), A=(46,0), E=(72,0), B=(46+13/2,13*3^.5/2);
+
real Radius = 4;
pair P=(C+D)/2, Q=(D+A)/2, R=(A+E)/2, T=(A+B)/2;
+
pair O=(0,0), A=(Radius,0), B=A*expi(pi/3), C=((63^.5-3^.5)/2,0), D=C*expi(pi/3);
draw(D--A--B--C--cycle);
+
draw(Circle(O,Radius));
draw(C--A);
+
dot(O);
draw(A--E--B,dashed);
+
draw(B--O); draw(C--A);
label("\(A\)",A,SSW);
+
pair F=(C+(0.5*3^0.5,0.5)), G=(D+(0.5*3^0.5,0.5));
label("\(B\)",B,NNE);
+
draw(C--D--G--F);
label("\(C\)",C,WNW);
+
draw(O--C--F--cycle,linewidth(1));
label("\(D\)",D,SSW);
+
picture p = new picture;  
label("\(E\)",E,SSE);
+
draw(p,Circle(C,0.15));
label("24\(\sqrt{3}\)",P,W);
+
clip(p,O--C--F--cycle);
label("46",Q,S);
+
add(p);
label("26",R,S);
+
p = new picture;  
label("13",T,WNW);
+
draw(p,Circle(D,0.15));
 +
clip(p,O--D--C--cycle);
 +
add(p);
 +
clip(currentpicture,B+(0,1)--O-(0.2,0.2)--A+(1,0)--cycle);
 +
label("\(x\)",(O+C)/2,S);
 +
label("\(x\)",(D+C)/2,NE);
 +
label("\(1\)",(F+C)/2,SE);
 +
label("\(4\)",(O+F)/2,NW);
 +
label("\(150^{\circ}\)",C+(-0.08,0.08),NW);
 +
label("60",D+(0,-0.15),S);
 
</asy></center>
 
</asy></center>
 +
 +
By the [[Law of Cosines]] on the bolded triangle,
 +
<cmath>\begin{align*}
 +
4^2 &= x^2 + 1^2 - 2 \cdot x \cdot 1 \cos 150^{\circ}\
 +
0 &= x^2 + \sqrt{3}x - 15 \
 +
x &= \boxed{\frac{3\sqrt{7}-\sqrt{3}}{2}}
 +
\end{align*}
 +
</cmath>

Revision as of 18:44, 18 February 2008

[asy] defaultpen(fontsize(11pt)); size(250); real Radius = 4; pair O=(0,0), A=(Radius,0), B=A*expi(pi/3), C=((63^.5-3^.5)/2,0), D=C*expi(pi/3); draw(Circle(O,Radius)); dot(O); draw(B--O); draw(C--A); pair F=(C+(0.5*3^0.5,0.5)), G=(D+(0.5*3^0.5,0.5)); draw(C--D--G--F); draw(O--C--F--cycle,linewidth(1)); picture p = new picture;  draw(p,Circle(C,0.15)); clip(p,O--C--F--cycle); add(p); p = new picture;  draw(p,Circle(D,0.15)); clip(p,O--D--C--cycle); add(p); clip(currentpicture,B+(0,1)--O-(0.2,0.2)--A+(1,0)--cycle); label("\(x\)",(O+C)/2,S); label("\(x\)",(D+C)/2,NE); label("\(1\)",(F+C)/2,SE); label("\(4\)",(O+F)/2,NW); label("\(150^{\circ}\)",C+(-0.08,0.08),NW); label("\(60^{\circ}\)",D+(0,-0.15),S); [/asy]

By the Law of Cosines on the bolded triangle, \begin{align*} 4^2 &= x^2 + 1^2 - 2 \cdot x \cdot 1 \cos 150^{\circ}\\ 0 &= x^2 + \sqrt{3}x - 15 \\ x &= \boxed{\frac{3\sqrt{7}-\sqrt{3}}{2}} \end{align*}