Difference between revisions of "User:Azjps/sandbox"
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m (asy testing) |
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<center><asy> | <center><asy> | ||
| − | + | defaultpen(fontsize(11pt)); | |
| − | + | size(250); | |
| − | pair | + | 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(D-- | + | draw(Circle(O,Radius)); |
| − | draw(C-- | + | dot(O); |
| − | draw( | + | 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)); | |
| − | label("\( | + | draw(C--D--G--F); |
| − | label("\( | + | draw(O--C--F--cycle,linewidth(1)); |
| − | label("\( | + | picture p = new picture; |
| − | label("\( | + | draw(p,Circle(C,0.15)); |
| − | label(" | + | clip(p,O--C--F--cycle); |
| − | + | add(p); | |
| − | label(" | + | 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></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 19:44, 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]](http://latex.artofproblemsolving.com/a/d/4/ad497431b55dfb56892013929c69be6651556b01.png)
By the Law of Cosines on the bolded triangle,
