In an infinite sequence of regular hexagons, each hexagon except the first is formed by connecting the midpoints of the sides of the previous hexagon. If the perimeter of the first hexagon is 12 and the sum of the perimeters of all the hexagons is a(b+root 3) , then find (a + b).
Answers
Given : In an infinite sequence of regular hexagons, each hexagon except the first is formed by connecting the midpoints of the sides of the previous hexagon. If the perimeter of the first hexagon is 12 and the sum of the perimeters of all the hexagons is a(b+root 3)
To find : (a + b).
Solution:
Perimeter of 1st hexagon is 12
hence each side = 12/6 = 2
internal angle in each hexagon = 120°
Length of side by connecting mid point = √(1² + 1² - 2(1)(1)Cos120°)
= √(1 +1 - 2(1)(1)(-0.5))
= √(2+1)
= √3
Perimeter = 6√3
Side = √3
Next side would be 3/2 hence perimeter = 9
12 + 6√3 + 9 +..................
r = √3/2
a = 12
S = a/(1 - r)
= 12/(1 - √3/2)
= 12/(2 - √3)
= 12 (2 + √3)
Comparing with
a(b+root 3)
a = 12
b = 2
a + b = 12 + 2 = 14
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