Calculus mathematic question

Question

Hassan has a piece of paper in the shape of a hexagon. The interior angles are all 120◦

, and

the side lengths are 1, 2, 3, 4, 5, 6, although not in that order. Initially, the paper is in

the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle

shapes, one at each corner of the paper. What is the minimum possible side length of the

original triangle?

Answer 1

If we start with an equilateral triangle, and chop off each of the 3 corners, we divide each side of the triangle into 3 unequal segments. Now we have the total of 9 segments. At least one of these segments is 6 units long. If we want to minimize the length of the original segment, the other two segments have to be 1 unit and 2 units long, and the resulting side length will be 1+2+6 = 9

To make the hexagon one needs to cut off three small equilateral triangles with side lengths of 1 unit, 2 units and 3 units.

Answer 2

forgoing angle confusion,

Want smallest original triangle equilateral

notice every other side of hexagon is a side created by chopping off equilateral triangle

choose chop off equilateral triangles to be 1,2,3 length from a unknown size equilateral triangle

chopping off smallest amount will provide minimum sized original triangle

draw and label lengths. Will find three unknown sides, label x,y,z

will find equation right off, x+y+z=15 since that is the sum of missing sides

on an adjoining side pair will have (I have in my diagram) x+4=y+5 (sides must be equal)

on next adjoining side pair have y+5=z+3 (In my diagram)

that is enough to find y=4 and side is 9.

Answer 3

interior angles of a regular polygon are 180(n-2)/n where n = number of sides

for a hexagon, 180(6-2)/6 = 120 degrees for each interior angle, but generally only if the sides are equal

if sides are 1,2,3,4,5 and 6, the 6 angles are generally all unequal, although their sum is always 720 degrees like a regular hexagon

there might be a problem with the problem, as written

nonetheless, a wild guess is 12 may be the answer you want,

with the hexagon sides in order 1,6,2,4,3,5, opposite sides parallel

that's from cutting off 3 equilateral triangles of one with side 6, another with side 5 and the 3rd with side 4

a smaller equilateral triangle is with side 9 will get a hexagon with the same order 1,6,2,4,3,5 but by cutting off equilateral triangles with one side 1, another with side 2 and 3rd with side 3

(1/3)(2(1)+2(2)+2(3) + 4 +5 +6) = (1/3(2+4+6+15) =(1/3)27 = 9

another possibility is take equilateral triangles with sides 1, 3 and 5

it creates a hexagon with sides in order 1,6,3,25,4, and sides 10 >9, but <12

12 is the largest, 9 the smallest, 10 in between