Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "User:Jonjoseph"
 
(16 intermediate revisions by the same user not shown)
Line 49: Line 49:
 
label("$T$",T,NE);
 
label("$T$",T,NE);
 
label("$S$",S,NW);
 
label("$S$",S,NW);
−
label("$Z$",Z,SW);
+
label("$Z$",Z,E);
 
label("$Y$",Y,SE);
 
label("$Y$",Y,SE);
 
label("$U$",U,E);
 
label("$U$",U,E);
Line 67: Line 67:
 
draw(Z--V,dashed);
 
draw(Z--V,dashed);
 
draw(W--U,red);
 
draw(W--U,red);
 +
</asy>
 +
 +
<asy>
 +
unitsize(1 cm);
 +
pair A, B, C;
 +
A = (0, 0);
 +
B = (2, 0);
 +
C = (2, 1.5);
 +
 +
draw(A--B--C--cycle);
 +
label("$A$", A, W);
 +
label("$B$", B, E);
 +
label("$C$", C, N);
 +
label("$5$", (A+B)/2, S);
 +
label("$6$", (B+C)/2, E);
 +
label("$\theta$", A, NE);
 +
</asy>
 +
 +
<asy>
 +
unitsize(0.2 cm);
 +
pair A, B, E, P;
 +
A = (0,0);
 +
B = (8,0);
 +
E=(1,2);
 +
P = (B + reflect(A,(1,2))*(B))/2;
 +
draw((-10,0)--(10,0));
 +
draw(A+(-1,-2)--8*E, red);
 +
draw(A--(0,25));
 +
draw(B+-1*(1,2)--(9, 2)+7*(1,2), red);
 +
draw(B--P,dashed);
 +
dot("$A$", A, SE);
 +
dot("$B$", B, SE);
 +
dot("$P$",  P, N);
 +
label("$8$", (A + B)/2, S);
 +
</asy>
 +
 +
<asy>
 +
size(125);
 +
real ticklen=3;
 +
real tickspace=2;
 +
 +
real ticklength=0.1cm;
 +
real axisarrowsize=0.14cm;
 +
pen axispen=black+1.3bp;
 +
real vectorarrowsize=0.2cm;
 +
real tickdown=-0.5;
 +
real tickdownlength=-0.15inch;
 +
real tickdownbase=0.3;
 +
real wholetickdown=tickdown;
 +
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
 +
    import graph;
 +
    real i;
 +
    if(complexplane) {
 +
        label("$\textnormal{Re}$",(xright,0),SE);
 +
        label("$\textnormal{Im}$",(0,ytop),NW);
 +
    } else {
 +
        label("$x$",(xright+0.4,-0.5));
 +
        label("$y$",(-0.5,ytop+0.2)); 
 +
    }
 +
 +
    ylimits(ybottom,ytop);
 +
    xlimits( xleft, xright);
 +
    real[] TicksArrx,TicksArry;
 +
 +
    for(i=xleft+xstep; i<xright; i+=xstep) {
 +
        if(abs(i) >0.1) {
 +
            TicksArrx.push(i);
 +
        }
 +
    }
 +
    for(i=ybottom+ystep; i<ytop; i+=ystep) {
 +
        if(abs(i) >0.1) {
 +
            TicksArry.push(i);
 +
        }
 +
    }
 +
 +
    if(usegrid) {
 +
        xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.72),extend=true),p=invisible);//,above=true);
 +
        yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.72),extend=true), p=invisible);//,Arrows);
 +
    }
 +
    if(useticks) {
 +
        xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry ,  pTick=black+0.8bp,Size=ticklength), above=true,  Arrows(size=axisarrowsize));
 +
        yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx ,  pTick=black+0.8bp,Size=ticklength), above=true,  Arrows(size=axisarrowsize));
 +
 +
    } else {
 +
        xequals(0, ymin=ybottom, ymax=ytop, p=axispen,  above=true,  Arrows(size=axisarrowsize));
 +
        yequals(0, xmin=xleft, xmax=xright, p=axispen,  above=true,  Arrows(size=axisarrowsize));
 +
    }
 +
};
 +
rr_cartesian_axes(-5,5,-1,10);
 +
real f(real x) {return x*x;}
 +
real g(real x) {return 2*x*x;}
 +
draw(graph(f,-sqrt(10),sqrt(10),operator ..), red ,Arrows(size=axisarrowsize));
 +
draw(graph(g,-sqrt(5),sqrt(5),operator ..), blue ,Arrows(size=axisarrowsize));
 +
 +
//dot((2,4), p=orange);
 +
//dot((2,8),p=orange);
 +
//draw((2,4)--(sqrt(2),4),p=orange, Arrow(size=2*axisarrowsize));
 +
draw((3,9)--(3/sqrt(2),9),p=black, Arrow(size=2*axisarrowsize));
 +
dot((3, 9), p=black);
 +
 +
//dot((-1,1), p=purple);
 +
//dot((-1,2),p=purple);
 +
//draw((-1,1)--(-1,2),p=purple, Arrow(size=2*axisarrowsize));
 +
</asy>
 +
 +
<asy>
 +
size(170);
 +
draw(circle((0,0),1));
 +
draw((0,0)--(1,0)--(1,.75)--cycle);
 +
draw((0.9,0)--(0.9,0.1)--(1,0.1));
 +
label("$\theta$", (0.15,0), NE);
 +
label("1", (0.5,0), S);
 +
label("$\tan \theta$", (1,.375), E);
 +
label("$\sec\theta$", (.5, .375), NW);
 +
</asy>
 +
 +
<asy>
 +
size(200);
 +
import three;
 +
currentprojection = orthographic(3.1, 2.1, 1);
 +
triple A, B, X, Y, Z;
 +
A = (1, 1, 0);
 +
B = (0, 0, 1);
 +
O = (0,0,0);
 +
X = (1, 0, 0);
 +
Y = (0, 1, 0);
 +
Z = (0, 0, 1);
 +
 +
real big = 2;
 +
 +
draw(surface(-(big - 0.1)*X - (big - 0.1)*Y -- -(big - 0.1)*X + (big - 0.1)*Y -- (big - 0.1)*X + (big - 0.1)*Y -- (big - 0.1)*X - (big - 0.1)*Y--cycle), rgb(226, 253, 255), nolight);
 +
draw(O--big*X, Arrow3(size = 0.2cm));
 +
draw(O-- -big*X, Arrow3(size = 0.2cm));
 +
draw(O-- big*Y, Arrow3(size = 0.2cm));
 +
draw(O-- -big*Y, Arrow3(size = 0.2cm));
 +
draw(O-- big*Z, Arrow3(size = 0.2cm));
 +
draw(O-- -big*Z, Arrow3(size = 0.2cm));
 +
 +
label("$x$", big*X, NW);
 +
label("$y$", big*Y, N);
 +
label("$z$", big*Z, W);
 +
 +
draw(A--B, red, Arrow3(size = 0.3cm));
 +
 +
dot("$A$", A, S);
 +
dot("$B$", B, E);
 +
</asy>
 +
 +
<asy>
 +
import graph;
 +
 +
//size(6cm);
 +
picture pic1;
 +
real xsize=200, ysize=140;
 +
size(pic1,xsize,ysize,IgnoreAspect);
 +
 +
real f(real x){return ((x - .5)^3 - .5 x + 1);}
 +
xlimits(pic1,  -2, 2);
 +
ylimits(pic1, -2,2);
 +
draw(pic1, graph(f,-1,2),black+1bp);
 +
xaxis(pic1, Label("$x$",position=EndPoint, align=SE),Ticks("%"), Arrow);
 +
yaxis(pic1, Label("$y$",position=EndPoint, align=NW),Ticks("%"), Arrow);
 +
labelx(pic1, 1,2S);
 +
labely(pic1, 1,2W);
 +
label(pic1, "\scriptsize $y = g(x)$", (2.2,1));
 +
 +
picture pic2;
 +
size(pic2,xsize,ysize,IgnoreAspect);
 +
 +
real g(real x){return (3(x - .5)^2 - .5);}
 +
xlimits(pic2,  -2, 2);
 +
ylimits(pic2, -2,2);
 +
draw(pic2, graph(g,-0.5,1.5),black+1bp);
 +
xaxis(pic2, Label("$x$",position=EndPoint, align=SE),Ticks("%"), Arrow);
 +
yaxis(pic2, Label("$y$",position=EndPoint, align=NW),Ticks("%"), Arrow);
 +
labelx(pic2, 1,2S);
 +
labely(pic2, 1,2W);
 +
label(pic2, "\scriptsize $y = g'(x)$", (2.2,1));
 +
 +
picture pic3;
 +
size(pic3,xsize,ysize,IgnoreAspect);
 +
 +
real h(real x){return (6(x - .5));}
 +
xlimits(pic3,  -2, 2);
 +
ylimits(pic3, -2,2);
 +
draw(pic3, graph(h,0.1,0.75),black+1bp);
 +
xaxis(pic3, Label("$x$",position=EndPoint, align=SE),Ticks("%"), Arrow);
 +
yaxis(pic3, Label("$y$",position=EndPoint, align=NW),Ticks("%"), Arrow);
 +
labelx(pic3, 1,2S);
 +
labely(pic3, 1,2W);
 +
label(pic3, "\scriptsize $y = g''(x)$", (2.2,1));
 +
 +
// Fit pic to W of origin:
 +
// add(pic.fit(),(0,0),W);
 +
 +
real margin = 5mm;
 +
add(pic1.fit(),(0,0), margin * N);
 +
 +
// Fit pic2 to E of (5mm,0):
 +
add(pic2.fit(),(0,0), margin * S);
 +
 +
add(pic3.fit(), (0, 0), 12*margin*S);
 
</asy>
 
</asy>

Latest revision as of 20:43, 28 March 2022

[asy] pair O = (0,0); pair P = (2,0); draw(unitcircle); dot(O); dot(P); label('$P$', P, E); label('$C$', O, W); [/asy]

Given a circle with center $C$ and a point $P$ that lies outside the circle, bisect circle $C$ with a second circle centered at $P$.

[asy] pair O = (0,0); pair P = (2,0); pair A = (0,1); pair B = (0, -1); draw(unitcircle); draw(A--B, red); draw(P--O, red); draw(circle(P, 2.24)); dot(O); dot(P); dot(A); dot(B); label('$P$', P, E); label('$C$', O, W); [/asy]

Problem from class:

[asy] import olympiad; import cse5; import geometry; size(150); defaultpen(fontsize(10pt)); defaultpen(0.8); dotfactor=4; pair X = origin; pair Y = (3,0); pair W = X+dir(120); pair Z = Y+dir(120); pair T = (0,2); pair S = T+dir(120); pair U = (3,2); pair V = U+dir(120); label("$X$",X,SW); label("$W$",W,SW); label("$T$",T,NE); label("$S$",S,NW); label("$Z$",Z,E); label("$Y$",Y,SE); label("$U$",U,E); label("$V$",V,NE); draw(X--Y); draw(Z--Y, dashed); draw(W--X, red); draw(X--Z, red); draw(Z--U, red); draw(Z--W,red); draw(U--X, red); draw(S--T--U--V--cycle); draw(X--T); draw(W--S); draw(Y--U); draw(Z--V,dashed); draw(W--U,red); [/asy]

[asy] unitsize(1 cm); pair A, B, C; A = (0, 0); B = (2, 0); C = (2, 1.5); draw(A--B--C--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$5$", (A+B)/2, S); label("$6$", (B+C)/2, E); label("$\theta$", A, NE); [/asy]

[asy] unitsize(0.2 cm); pair A, B, E, P; A = (0,0); B = (8,0); E=(1,2); P = (B + reflect(A,(1,2))*(B))/2; draw((-10,0)--(10,0)); draw(A+(-1,-2)--8*E, red); draw(A--(0,25)); draw(B+-1*(1,2)--(9, 2)+7*(1,2), red); draw(B--P,dashed); dot("$A$", A, SE); dot("$B$", B, SE); dot("$P$", P, N); label("$8$", (A + B)/2, S); [/asy]

[asy] size(125); real ticklen=3; real tickspace=2; real ticklength=0.1cm; real axisarrowsize=0.14cm; pen axispen=black+1.3bp; real vectorarrowsize=0.2cm; real tickdown=-0.5; real tickdownlength=-0.15inch; real tickdownbase=0.3; real wholetickdown=tickdown; void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) { import graph; real i; if(complexplane) { label("$\textnormal{Re}$",(xright,0),SE); label("$\textnormal{Im}$",(0,ytop),NW); } else { label("$x$",(xright+0.4,-0.5)); label("$y$",(-0.5,ytop+0.2)); } ylimits(ybottom,ytop); xlimits( xleft, xright); real[] TicksArrx,TicksArry; for(i=xleft+xstep; i<xright; i+=xstep) { if(abs(i) >0.1) { TicksArrx.push(i); } } for(i=ybottom+ystep; i<ytop; i+=ystep) { if(abs(i) >0.1) { TicksArry.push(i); } } if(usegrid) { xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.72),extend=true),p=invisible);//,above=true); yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.72),extend=true), p=invisible);//,Arrows); } if(useticks) { xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize)); yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize)); } else { xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize)); yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize)); } }; rr_cartesian_axes(-5,5,-1,10); real f(real x) {return x*x;} real g(real x) {return 2*x*x;} draw(graph(f,-sqrt(10),sqrt(10),operator ..), red ,Arrows(size=axisarrowsize)); draw(graph(g,-sqrt(5),sqrt(5),operator ..), blue ,Arrows(size=axisarrowsize)); //dot((2,4), p=orange); //dot((2,8),p=orange); //draw((2,4)--(sqrt(2),4),p=orange, Arrow(size=2*axisarrowsize)); draw((3,9)--(3/sqrt(2),9),p=black, Arrow(size=2*axisarrowsize)); dot((3, 9), p=black); //dot((-1,1), p=purple); //dot((-1,2),p=purple); //draw((-1,1)--(-1,2),p=purple, Arrow(size=2*axisarrowsize)); [/asy]

[asy] size(170); draw(circle((0,0),1)); draw((0,0)--(1,0)--(1,.75)--cycle); draw((0.9,0)--(0.9,0.1)--(1,0.1)); label("$\theta$", (0.15,0), NE); label("1", (0.5,0), S); label("$\tan \theta$", (1,.375), E); label("$\sec\theta$", (.5, .375), NW); [/asy]

[asy] size(200); import three; currentprojection = orthographic(3.1, 2.1, 1); triple A, B, X, Y, Z; A = (1, 1, 0); B = (0, 0, 1); O = (0,0,0); X = (1, 0, 0); Y = (0, 1, 0); Z = (0, 0, 1); real big = 2; draw(surface(-(big - 0.1)*X - (big - 0.1)*Y -- -(big - 0.1)*X + (big - 0.1)*Y -- (big - 0.1)*X + (big - 0.1)*Y -- (big - 0.1)*X - (big - 0.1)*Y--cycle), rgb(226, 253, 255), nolight); draw(O--big*X, Arrow3(size = 0.2cm)); draw(O-- -big*X, Arrow3(size = 0.2cm)); draw(O-- big*Y, Arrow3(size = 0.2cm)); draw(O-- -big*Y, Arrow3(size = 0.2cm)); draw(O-- big*Z, Arrow3(size = 0.2cm)); draw(O-- -big*Z, Arrow3(size = 0.2cm)); label("$x$", big*X, NW); label("$y$", big*Y, N); label("$z$", big*Z, W); draw(A--B, red, Arrow3(size = 0.3cm)); dot("$A$", A, S); dot("$B$", B, E); [/asy]

[asy] import graph; //size(6cm); picture pic1; real xsize=200, ysize=140; size(pic1,xsize,ysize,IgnoreAspect); real f(real x){return ((x - .5)^3 - .5 x + 1);} xlimits(pic1, -2, 2); ylimits(pic1, -2,2); draw(pic1, graph(f,-1,2),black+1bp); xaxis(pic1, Label("$x$",position=EndPoint, align=SE),Ticks("%"), Arrow); yaxis(pic1, Label("$y$",position=EndPoint, align=NW),Ticks("%"), Arrow); labelx(pic1, 1,2S); labely(pic1, 1,2W); label(pic1, "\scriptsize $y = g(x)$", (2.2,1)); picture pic2; size(pic2,xsize,ysize,IgnoreAspect); real g(real x){return (3(x - .5)^2 - .5);} xlimits(pic2, -2, 2); ylimits(pic2, -2,2); draw(pic2, graph(g,-0.5,1.5),black+1bp); xaxis(pic2, Label("$x$",position=EndPoint, align=SE),Ticks("%"), Arrow); yaxis(pic2, Label("$y$",position=EndPoint, align=NW),Ticks("%"), Arrow); labelx(pic2, 1,2S); labely(pic2, 1,2W); label(pic2, "\scriptsize $y = g'(x)$", (2.2,1)); picture pic3; size(pic3,xsize,ysize,IgnoreAspect); real h(real x){return (6(x - .5));} xlimits(pic3, -2, 2); ylimits(pic3, -2,2); draw(pic3, graph(h,0.1,0.75),black+1bp); xaxis(pic3, Label("$x$",position=EndPoint, align=SE),Ticks("%"), Arrow); yaxis(pic3, Label("$y$",position=EndPoint, align=NW),Ticks("%"), Arrow); labelx(pic3, 1,2S); labely(pic3, 1,2W); label(pic3, "\scriptsize $y = g''(x)$", (2.2,1)); // Fit pic to W of origin: // add(pic.fit(),(0,0),W); real margin = 5mm; add(pic1.fit(),(0,0), margin * N); // Fit pic2 to E of (5mm,0): add(pic2.fit(),(0,0), margin * S); add(pic3.fit(), (0, 0), 12*margin*S); [/asy]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User:Jonjoseph&oldid=173037"