Square into three rectangles

Divide the square with a side length of 12 cm into three rectangles with the same circumference so that these circumferences are as small as possible.

Final Answer:

a =  12
b =  3
c =  9
d =  6
o1 =  30
o2 =  30
o3 =  30
o4 =  32

Step-by-step explanation:


a=12
o1 = 2(a+b)
o2 = 2(c+d)
o3 = 2(c+d)
o1=o2
b+c=a
2d = 12
o4 = 2(a+a/3)


a=12
o1 = 2·(a+b)
o2 = 2·(c+d)
o3 = 2·(c+d)
o1=o2
b+c=a
2·d = 12
o4 = 2·(a+a/3)

a = 12
2a+2b-o1 = 0
2c+2d-o2 = 0
2c+2d-o3 = 0
o1-o2 = 0
a-b-c = 0
2d = 12
8a-3o4 = 0

Pivot: Row 1 ↔ Row 8
8a-3o4 = 0
2a+2b-o1 = 0
2c+2d-o2 = 0
2c+2d-o3 = 0
o1-o2 = 0
a-b-c = 0
2d = 12
a = 12

Row 2 - 2/8 · Row 1 → Row 2
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2c+2d-o3 = 0
o1-o2 = 0
a-b-c = 0
2d = 12
a = 12

Row 6 - 1/8 · Row 1 → Row 6
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2c+2d-o3 = 0
o1-o2 = 0
-b-c+0.375o4 = 0
2d = 12
a = 12

Row 8 - 1/8 · Row 1 → Row 8
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2c+2d-o3 = 0
o1-o2 = 0
-b-c+0.375o4 = 0
2d = 12
0.375o4 = 12

Row 6 - -1/2 · Row 2 → Row 6
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2c+2d-o3 = 0
o1-o2 = 0
-c-0.5o1+0.75o4 = 0
2d = 12
0.375o4 = 12

Row 4 - Row 3 → Row 4
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
o2-o3 = 0
o1-o2 = 0
-c-0.5o1+0.75o4 = 0
2d = 12
0.375o4 = 12

Row 6 - -1/2 · Row 3 → Row 6
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
o2-o3 = 0
o1-o2 = 0
d-0.5o1-0.5o2+0.75o4 = 0
2d = 12
0.375o4 = 12

Pivot: Row 4 ↔ Row 7
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2d = 12
o1-o2 = 0
d-0.5o1-0.5o2+0.75o4 = 0
o2-o3 = 0
0.375o4 = 12

Row 6 - 1/2 · Row 4 → Row 6
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2d = 12
o1-o2 = 0
-0.5o1-0.5o2+0.75o4 = -6
o2-o3 = 0
0.375o4 = 12

Row 6 - -0.5 · Row 5 → Row 6
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2d = 12
o1-o2 = 0
-o2+0.75o4 = -6
o2-o3 = 0
0.375o4 = 12

Row 7 + Row 6 → Row 7
8a-3o4 = 0
2b-o1+0.75o4 = 0
2c+2d-o2 = 0
2d = 12
o1-o2 = 0
-o2+0.75o4 = -6
-o3+0.75o4 = -6
0.375o4 = 12


o4 = 12/0.375 = 32
o3 = -6-0.75o4/-1 = -6-0.75 · 32/-1 = 30
o2 = -6-0.75o4/-1 = -6-0.75 · 32/-1 = 30
o1 = 0+o2 = 0+30 = 30
d = 12/2 = 6
c = 0-2d+o2/2 = 0-2 · 6+30/2 = 9
b = 0+o1-0.75o4/2 = 0+30-0.75 · 32/2 = 3
a = 0+3o4/8 = 0+3 · 32/8 = 12

a = 12
b = 3
c = 9
d = 6
o1 = 30
o2 = 30
o3 = 30
o4 = 32

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Showing 1 comment:
Dr Math
so the solution is

1. 12*3 ...  a = 12 * b = 3
2. 9*6 ... c = 9 * d = 6
3. 9*6 ... c = 9 * d = 6

o1 is the circumference of the first, o2 the second, o3 the third rectangle
o4 is a circumference if the square is divided into same   12 * 4 rectangles





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