Qual é a área que falta neste hexágono regular?
(What is the missing area in this regular hexagon?)
PCFilho
(pescado no Twitter da professora Catriona Agg)
quarta-feira, 10 de março de 2021
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Are the 1 and 2 in the figure simply labels, or are they the areas of those regions?
ResponderExcluirThey are the areas of the regions.
ExcluirOK. Here's my answer.
ExcluirGiven: The hexagon is regular and the numbers in the regions are the areas of those regions.
Question: What is the area of the light green region?
First, we define a few variables:
s = side length of hexagon
x = length of shortest side of yellow region
Since the hexagon is regular, each of its interior angles has a measure of [(6 – 2)/6]180° = 120°. The yellow region is a triangle. This triangle has side lengths of s and x, and the measure of the angle between them is 120°, so its area is (1/2)(s)(x)(sin 120°) = (1/2)(s)(x)(√3/2) = (√3/4)sx. Since this area is equal to 1, we have sx = 4/√3.
To calculate the area of the dark green region in terms of s and x, we draw a vertical line between the upper left and lower left vertices of the hexagon. This divides the region into two triangular regions, one of which is right and the other of which is isosceles. The right triangle’s base is equal to (s – x) and its height is s√3, while the isosceles triangle has two sides each having a length equal to s and the angle between those sides has a measure of 120°. So, the area of the dark green region is (1/2)(s – x)(s√3) + (1/2)(s)(s)(√3/2) = (√3/4)(3s² – 2sx) = (3√3/4)s² – (√3/2)sx. This area is equal to 2, and sx = 4/√3, so:
(3√3/4)s² – (√3/2)(4/√3) = 2
(3√3/4)s² – 2 = 2
(3√3/4)s² = 4
s² = 16/(3√3)
We could solve for s here, but we don’t need to. We use this value to find the area of the hexagon:
Area of hexagon = (3√3/2)s² = (3√3/2)[16/(3√3)] = 8
Therefore:
Area of light green region = area of hexagon – (area of yellow region + area of dark green region)
= 8 – (1 + 2)
= 5
My answer: 5 square units
P.S.: In case anyone’s wondering, the value of s is √[16/(3√3)] = 1.755 and the value of x is √[(16/3)(3√3/16)] = √(√3) = 1.316. As expected, s > x.
Well done, Jake!!
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