Two identical regular hexagons are inscribed in a rectangle whose area is 2563 square inches. What is the perimeter of each hexagon in inches?a)48b)56c)64d)563e)643Correct answer is option 'A'. Can you explain this answer?

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Two identical regular hexagons are inscribed in a rectangle whose area is 256√3 square inches. What is the perimeter of each hexagon in inches?

a)
48
b)
56
c)
64
d)
56√3
e)
64√3

Correct answer is option 'A'. Can you explain this answer?

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Two identical regular hexagons are inscribed in a rectangle whose area...

To find – Perimeter of hexagon

Given –

The two hexagons are identical – the length of the sides are equal.
Area of rectangle = 256√3 in²

Approach –

Let ‘a’ be the side of the hexagon and let ‘x’ be the length of the rectangle.
To find the perimeter of the hexagon, we need to express the sides of rectangle (x) in terms of the side of the hexagon (a).
Then given the area we can find the measure of ‘a’
Hence, we can find the perimeter.

Solution

Each side of the hexagon is ‘a’ since it is a regular hexagon.
Consider the middle shaded triangle.
It is an equilateral triangle since each angle is 60° as explained below.
- Interior angle of the hexagon = 120°.
- The angle made by AB = 180°.
- Thus, angle of the triangle = 180 – 120 = 60°
- The purple line is a/2.
- x/2 = a/2+a+a/2 = 2a
- x = 4a

Likewise, applying Pythagoras theorem on the shaded side triangle we get the value of y in terms of a

a²-a²/4 = y²/4
y = a√3

We can now substitute the values of x and y in terms of a in the equation for area of the rectangle.

xy = 256√3
4a²√3 = 256√3
a = 8
Perimeter of hexagon = 8 * 6 = 48 inches

Correct Answer = Choice A

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Two identical regular hexagons are inscribed in a rectangle whose area is 256√3 square inches. What is the perimeter of each hexagon in inches?a)48b)56c)64d)56√3e)64√3Correct answer is option 'A'. Can you explain this answer?

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