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MATH / A hexagon is drawn with its vertices at (0,0),(1,0),(2,1),(2,2),(1,2), and (0,1), and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into 24 reg…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
A hexagon is drawn with its vertices at and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into regions of various shapes and sizes. These regions are shown in pink and yellow below. If the smallest region (by area) has area , and the largest has area , then what is the ratio ? Give your answer in lowest terms. [asy]
pair a=(0,0); pair b=(1,0); pair c=(2,1); pair d=(2,2); pair e=(1,2); pair f=(0,1);
pair g=(1,1);
pair h=(a+g)/2; pair i=(2*h+b)/3; pair j=(b+g)/2; pair k=(2*j+c)/3; pair l=(c+g)/2; pair m=(2*l+d)/3;
pair n=2*g-h; pair o=2*g-i; pair p=2*g-j; pair q=2*g-k; pair r=2*g-l; pair s=2*g-m;
fill(a--h--i--cycle,pink);
fill(a--i--b--cycle,yellow);
fill(i--b--j--cycle,pink);
fill(j--b--k--cycle,yellow);
fill(b--k--c--cycle,pink);
fill(k--c--l--cycle,yellow);
fill(l--c--m--cycle,pink);
fill(m--c--d--cycle,yellow);
fill(m--d--n--cycle,pink);
fill(n--d--o--cycle,yellow);
fill(o--d--e--cycle,pink);
fill(o--e--p--cycle,yellow);
fill(p--e--q--cycle,pink);
fill(q--e--f--cycle,yellow);
fill(f--r--q--cycle,pink);
fill(f--r--s--cycle,yellow);
fill(f--s--a--cycle,pink);
fill(a--s--h--cycle,yellow);
fill(g--r--q--p--cycle,yellow);
fill(g--p--o--n--cycle,pink);
fill(g--n--m--l--cycle,yellow);
fill(g--l--k--j--cycle,pink);
fill(g--j--i--h--cycle,yellow);
fill(g--h--s--r--cycle,pink);
draw(a--b--c--d--e--f--a,black+2);
draw(a--c--e--a);
draw(b--d--f--b);
draw(a--d);
draw(b--e);
draw(c--f);
[/asy]
Plain-text mathematical notation (without MathML)
A hexagon is drawn with its vertices at (0,0),(1,0),(2,1),(2,2),(1,2), and (0,1), and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into 24 regions of various shapes and sizes. These 24 regions are shown in pink and yellow below. If the smallest region (by area) has area a, and the largest has area b, then what is the ratio a:b? Give your answer in lowest terms. [asy] pair a=(0,0); pair b=(1,0); pair c=(2,1); pair d=(2,2); pair e=(1,2); pair f=(0,1); pair g=(1,1); pair h=(a+g)/2; pair i=(2*h+b)/3; pair j=(b+g)/2; pair k=(2*j+c)/3; pair l=(c+g)/2; pair m=(2*l+d)/3; pair n=2*g-h; pair o=2*g-i; pair p=2*g-j; pair q=2*g-k; pair r=2*g-l; pair s=2*g-m; fill(a--h--i--cycle,pink); fill(a--i--b--cycle,yellow); fill(i--b--j--cycle,pink); fill(j--b--k--cycle,yellow); fill(b--k--c--cycle,pink); fill(k--c--l--cycle,yellow); fill(l--c--m--cycle,pink); fill(m--c--d--cycle,yellow); fill(m--d--n--cycle,pink); fill(n--d--o--cycle,yellow); fill(o--d--e--cycle,pink); fill(o--e--p--cycle,yellow); fill(p--e--q--cycle,pink); fill(q--e--f--cycle,yellow); fill(f--r--q--cycle,pink); fill(f--r--s--cycle,yellow); fill(f--s--a--cycle,pink); fill(a--s--h--cycle,yellow); fill(g--r--q--p--cycle,yellow); fill(g--p--o--n--cycle,pink); fill(g--n--m--l--cycle,yellow); fill(g--l--k--j--cycle,pink); fill(g--j--i--h--cycle,yellow); fill(g--h--s--r--cycle,pink); draw(a--b--c--d--e--f--a,black+2); draw(a--c--e--a); draw(b--d--f--b); draw(a--d); draw(b--e); draw(c--f); [/asy]
Original LaTeX notation
A hexagon is drawn with its vertices at $$(0,0),(1,0),(2,1),(2,2),(1,2), \text{ and } (0,1),$$ and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into $24$ regions of various shapes and sizes. These $24$ regions are shown in pink and yellow below. If the smallest region (by area) has area $a$, and the largest has area $b$, then what is the ratio $a:b$? Give your answer in lowest terms. [asy]
pair a=(0,0); pair b=(1,0); pair c=(2,1); pair d=(2,2); pair e=(1,2); pair f=(0,1);
pair g=(1,1);
pair h=(a+g)/2; pair i=(2*h+b)/3; pair j=(b+g)/2; pair k=(2*j+c)/3; pair l=(c+g)/2; pair m=(2*l+d)/3;
pair n=2*g-h; pair o=2*g-i; pair p=2*g-j; pair q=2*g-k; pair r=2*g-l; pair s=2*g-m;
fill(a--h--i--cycle,pink);
fill(a--i--b--cycle,yellow);
fill(i--b--j--cycle,pink);
fill(j--b--k--cycle,yellow);
fill(b--k--c--cycle,pink);
fill(k--c--l--cycle,yellow);
fill(l--c--m--cycle,pink);
fill(m--c--d--cycle,yellow);
fill(m--d--n--cycle,pink);
fill(n--d--o--cycle,yellow);
fill(o--d--e--cycle,pink);
fill(o--e--p--cycle,yellow);
fill(p--e--q--cycle,pink);
fill(q--e--f--cycle,yellow);
fill(f--r--q--cycle,pink);
fill(f--r--s--cycle,yellow);
fill(f--s--a--cycle,pink);
fill(a--s--h--cycle,yellow);
fill(g--r--q--p--cycle,yellow);
fill(g--p--o--n--cycle,pink);
fill(g--n--m--l--cycle,yellow);
fill(g--l--k--j--cycle,pink);
fill(g--j--i--h--cycle,yellow);
fill(g--h--s--r--cycle,pink);
draw(a--b--c--d--e--f--a,black+2);
draw(a--c--e--a);
draw(b--d--f--b);
draw(a--d);
draw(b--e);
draw(c--f);
[/asy]level
Level 5
type
Geometry
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