```asy
size(7cm);
pair O,A,B,C;
O=(0,0);
A=(5,0);
B=(0,1);
C=(1,1.5);
draw(O--A--A+B--B--cycle);
draw(A--A+C--A+B+C--B+C--B);
draw(A+B--A+B+C);
label("$x
quot;,midpoint(B--B+C),WNW);
label("$yquot;,midpoint(B+C--A+B+C),N);
label("$zquot;,midpoint(A+B+C--A+C),E);
```
![[Chung-2018-t1-yc-p29.svg]]
---
```asy
import geometry;
size(7cm);
pair O;
O=(0,0);
perpendicular(O, NE, (1,0));
draw(arc(O, 1, 90, 360)--O--cycle);
label("$Oquot;, O, SW);
label("$rquot;, (0,0.5), W);
label("$rquot;, (0.5,0), S);
```
![[Chung-2018-t1-yc-p30.svg]]
---
```asy
import geometry;
size(10cm);
add(grid(9,12,mediumgrey+linewidth(0.5)+dashed));
draw((0,5)--(9,5),Arrow());
draw((4,0)--(4,12),Arrow());
label("$Oquot;,(4,5),SW);
label("$f(x)quot;,(7,2),NE, Fill(white));
label("$3quot;,(4,8),E, Fill(white));
label("$3quot;,(7,5),S, Fill(white));
label("$-3quot;,(4,2),E, Fill(white));
label("$-3quot;,(1,5),S, Fill(white));
draw((2,12)--(8,0));
label("$xquot;,(9,5),E);
label("$yquot;,(4,12),N);
```
![[Chung-2018-t1-yc-p33.svg]]
---
```asy
import geometry;
size(10cm);
pair O=(0,0);
real r=5; real h=10;
draw(yscale(1/5)*arc(O,r,0,180),dashed);
draw(shift(O+(0,h))*yscale(1/5)*arc(O,r,0,180)--shift((0,0))*yscale(1/5)*arc(O,r,180,360)--cycle);
draw(shift(O+(0,h))*yscale(1/5)*arc(O,r,180,360));
label("$10quot;,(r,h/2),E);
draw(shift(O+(0,h))*yscale(1/5)*(O--r*(cos(pi/3),sin(pi/3))));
label("$rquot;,midpoint(shift(O+(0,h))*yscale(1/5)*(O--r*(cos(pi/3),sin(pi/3)))),SE);
pair O=(20,0);
real r=10; real h=2.5;
draw(shift((0,0))*yscale(1/5)*arc(O,r,0,180),dashed);
draw(shift((0,h))*yscale(1/5)*arc(O,r,0,180)--shift((0,0))*yscale(1/5)*arc(O,r,180,360)--cycle);
draw(shift((0,h))*yscale(1/5)*arc(O,r,180,360));
label("$hquot;,O+(r,h/2),E);
draw(shift((0,h))*yscale(1/5)*(O--O+r*(cos(pi/3),sin(pi/3))));
label("$2rquot;,midpoint(shift((0,h))*yscale(1/5)*(O--O+r*(cos(pi/3),sin(pi/3)))),SE);
```
![[Chung-2018-t1-yc-p35.svg]]
---
```
size(10cm);
import geometry;
import math;
pair P1,P2,P3,P4,P5,PT;
P1= (0,1);
P2= rotate(72)*P1;
P3= rotate(72)*P2;
P4= rotate(72)*P3;
P5= rotate(72)*P4;
PT= intersectionpoint(P1--P4,P5--P3);
draw(P1--P2--P3--P4--P5--cycle,dotted);
draw(P1--P3--P5--P4--cycle);
markangle(Label("$72^\circquot;,Relative(0.5)),n=1,radius=30,P1,P5,P3,heavyred);
markangle(Label("$72^\circquot;,Relative(0.5)),n=1,radius=-30,P1,PT,P5,heavyred);
markangle(Label("$72^\circquot;,Relative(0.7)),n=1,radius=-30,P1,P3,P4,heavyred);
markangle(Label("$72^\circquot;,Relative(0.7)),n=1,radius=30,P1,P4,P3,heavyred);
draw(circle((0,0),1));
label("$1quot;,(P1+P5)/2,NE);
label("$1quot;,(P1+PT)/2,WSW);
label("$1quot;,(P4+P5)/2,ESE);
label("$\varphiquot;,(P1+P3)/2,WNW);
label("$\varphi-1quot;,(PT+P4)/2,WSW);
```
![[golden-ratio-pentagon-diagonals.svg]]
---
More Special Right Triangles
![[special-right-triangles-in-regular-pentagon.svg]]
```
size(10cm);
import geometry;
import math;
pair P1,P2,P3,P4,P5;
P1= (0,1);
P2= rotate(72)*P1;
P3= rotate(72)*P2;
P4= rotate(72)*P3;
P5= rotate(72)*P4;
draw(P1--P2--P3--P4--P5--cycle,dotted);
markangle(Label("$72^\circquot;,Relative(0.3)),n=1,radius=-30,P1,P3,P4,heavyred);
markangle(Label("$18^\circquot;,Relative(0.45)),n=1,radius=-50,(P3+P4)/2,P1,P3,heavycyan);
draw(P1--(P3+P4)/2--P3--cycle);
label("$\frac{1+\sqrt5}2quot;,(P1+P3)/2,W);
label("$\frac12quot;,(P4+3*P3)/4,S);
markangle(Label("$36^\circquot;,Relative(0.5)),n=1,radius=30,P1,P5,P2,heavygreen);
markangle(Label("$54^\circquot;,Relative(0.45)),n=1,radius=30,(P3+P4)/2,P1,P5,purple);
draw(P5--(P5+P2)/2--P1--cycle);
label("$\frac{1+\sqrt5}4quot;,(3*P5+P2)/4,S);
label("$1quot;,(P1+P5)/2,NE);
label(scale(0.8)*"$\sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}quot;,(2.2*P1+P3+P4),E);
label(scale(0.8)*"$\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}}quot;,(1.4*P1+P3+P4),W);
draw(circle((0,0),1));
```
---
```
// This is my graph, there are many like it, but this one is mine.
size(8cm); import math; import graph;
real xmin, xmax, ymin, ymax, tl, wg;
xmin=-4; xmax=7;
ymin=-3; ymax=6;
wg=1; // Want grid? 1=yes 0=no
tl=0; //
pen gp = dashed+grey; // grid pen
// Grid
for (int i=ceil(xmin); i<=xmax; ++i) {draw((i,wg*ymin)--(i,wg*ymax),gp);}
for (int j=ceil(ymin); j<=ymax; ++j) {draw((wg*xmin,j)--(wg*xmax,j),gp);}
//Axes
draw((xmin,0)--(xmax,0),Arrow());
draw((0,ymin)--(0,ymax),Arrow());
// Short ticks
// for (int i=ceil(xmin); i<xmax; ++i) {draw((i,-tl)--(i,tl));}
// for (int j=ceil(ymin); j<ymax; ++j) {draw((-tl,j)--(tl,j));}
// Axes Labels
label("$xquot;,(xmax,0),E);
label("$yquot;,(0,ymax),N);
// Axes Labels
label("Hours",((xmax+xmin)/2,-1));
label(rotate(90)*minipage("\center{Number of Bacteria \\ (Thousands)}",width=5cm),((-3,ymax+ymin)/2));
//example of drawing on graph
draw((-3,3)--(0,0)--(1,1)--(3,-2)--(6,5));
label("$f(x)quot;,(6,5),SE,Fill(white+opacity(1)));
label("$5quot;,(5,0),S,Fill(white+opacity(1)));
label("$5quot;,(0,5),W,Fill(white+opacity(1)));
label("$1quot;,(1,0),S,Fill(white+opacity(1)));
label("$1quot;,(0,1),W,Fill(white+opacity(1)));
```
---
```
size(10cm); import math;
pair a1,a2,b1,b2,c1,c2,d1,d2,e1,e2;
real d = 0.15;
a1= exp((0,1)*(pi/2-d));
a2= exp((0,1)*(pi/2+d));
b1= exp((0,1)*(pi/2+2pi/5-d));
b2= exp((0,1)*(pi/2+2pi/5+d));
c1= exp((0,1)*(pi/2+4pi/5-d));
c2= exp((0,1)*(pi/2+4pi/5+d));
d1= exp((0,1)*(pi/2-4pi/5-d));
d2= exp((0,1)*(pi/2-4pi/5+d));
e1= exp((0,1)*(pi/2-2pi/5-d));
e2= exp((0,1)*(pi/2-2pi/5+d));
dot(a1); label("$0quot;,a1,NNE);
dot(a2); label("$8quot;,a2,NNW);
dot(b1); label("$1quot;,b1,NW);
dot(b2); label("$7quot;,b2,W);
dot(c1); label("$2quot;,c1,SW);
dot(c2); label("$6quot;,c2,S);
dot(d1); label("$3quot;,d1,SW);
dot(d2); label("$5quot;,d2,S);
dot(e1); label("$4quot;,e1,E);
dot(e2); label("$4quot;,e2,NE);
draw(a2--b1,dotted);
draw(a2--b2,dotted);
draw(a2--c1,dotted);
draw(a2--c2,dotted);
draw(a2--d1,dotted);
draw(a2--d2,dotted);
draw(a2--e1,dotted);
draw(a2--e2,dotted);
draw(b2--c1,dotted);
draw(b2--c2,dotted);
draw(b2--d1,dotted);
draw(b2--d2,dotted);
draw(b2--e1,dotted);
draw(b2--e2,dotted);
draw(c2--d1,dotted);
draw(c2--d2,dotted);
draw(c2--e1,dotted);
draw(c2--e2,dotted);
draw(d2--e1,dotted);
draw(d2--e2,dotted);
```
![[hcssim-p7.svg]]
---
selfmediant triangle
```
size(6cm);
pair A,B,C;
A=(0,0); B=(17*cos(44.758),17*sin(44.758)); C=(7,0);
draw(A--B--C--cycle);
path mA, mB, mC;
mA = A--(B+C)/2;
mB = B--(A+C)/2;
mC = C--(B+A)/2;
draw(mA,dashed+red);
draw(mB,dashed+heavygreen);
draw(mC,dashed+mediumblue);
draw(shift((10,0)+(C-A)/2)*mA,red);
draw(shift((10,0))*mB,heavygreen);
draw(shift((10,0)+(B-A)/2)*mC,mediumblue);
draw(((C/2)+B)/2--((C/2)+B)/2+(10,0),heavygreen+1+dotted+opacity(0.6),Arrow(3));
draw((C+B)/4--(3C+B)/4+(10,0),red+1+dotted+opacity(0.6),Arrow(3));
draw((2C+B)/4--(4C+6B)/8+(10,0),mediumblue+1+dotted+opacity(0.6),Arrow(3));
```
---
```
size(8cm); import math; import graph;
real xmin, xmax, ymin, ymax, tl, wg, wn;
xmin=0; xmax=8;
ymin=0; ymax=6;
wg=1; // Want grid? 1=yes 0=no
pen gp = dashed+grey; // grid pen
// Grid
for (int i=ceil(xmin); i<=xmax; ++i) {
draw(yscale(1.2)*(i,wg*ymin)--yscale(1.2)*(i,wg*ymax),gp);
if (i%2==0) {label("quot;+string(i)+"quot;,(i,0),S);}
}
for (int j=ceil(ymin); j<=ymax; ++j) {draw(yscale(1.2)*(wg*xmin,j)--yscale(1.2)*(wg*xmax,j),gp); label("quot;+string(80*j)+"quot;,yscale(1.2)*(0,j),W);
}
//Axes
draw((xmin,0)--(xmax,0));
draw(yscale(1.2)*(0,ymin)--yscale(1.2)*(0,ymax));
// Axes Labels
label("Hours",((xmax+xmin)/2,ymin-0.6),S);
label(rotate(90)*minipage("\center{Traveled distance \\ (in Miles)}",width=5cm),yscale(1.2)*((ceil(xmin)-2,ymax+ymin)/2),W);
path p = (0,0)..(1,1)..(3,2)..(4,3)..(6,3.5)..(7,5)..(8,6);
draw(yscale(1.2)*p);
```
![[workspace-59.svg]]
---
```
size(6cm);
real x, y;
x=6; y=20/x;
fill((0,y)--(-x,y)--(x,-y)--(x,0)--cycle,mediumgrey);
draw((x,y)--(-x,y)--(-x,-y)--(x,-y)--cycle^^(-x,y)--(x,-y)^^(x,0)--(0,y));
label("$Aquot;,(-x,-y),SW);
label("$Bquot;,(-x,y),NW);
label("$Cquot;,(x,y),NE);
label("$Dquot;,(x,-y),SE);
label("$Equot;,(0,y),N);
label("$Fquot;,(x,0),E);
```
![[workspace-60.svg]]
---
Kite
```
size(7cm);
real t=1; pair A,B,C,D;
A=(-t,0);
B=(0,t);
C=(2t,0);
D=(0,-t);
draw(A--B--C--D--cycle);
draw(A--C^^B--D);
draw((t/8,0)--(t/8,t/8)--(0,t/8));
label("$Aquot;,A,W);
label("$Bquot;,B,N);
label("$Cquot;,C,E);
label("$Dquot;,D,S);
```
![[1991-Mar-Reg-Geometry-p7.svg]]