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a square is constructed on each side of an equilateral triangle, and segments are drawn between adjacent external vertices of the three squares to form hexagon ABCDEF as shown. If the equilateral triangle has sides of length √3 - 1 units, what is the perimeter of the hexagon ABCDEF?

a square is constructed on each side of an equilateral triangle, and segments are-example-1
User Kris Braun
by
6.6k points

1 Answer

7 votes

Answer:


P=6\ units

Explanation:

we know that

The perimeter of the hexagon is equal to


P=AB+BC+CD+DE+EF+FA

Remember that


AB+CD=EF=(√(3)-1)\ units


BC=DE=FA ----> is the length base of an isosceles triangle

so

The perimeter is equal to


P=3(√(3)-1)+3BC

Find the length side BC

Let

x ----> the measure of the vertex angle in the isosceles triangle

The measure of the vertex angle in the isosceles triangle is equal to


90^o+60^o+90^o+x=360^o


x=120^o

Applying trigonometric property

In the right triangle of the attached figure


cos(30^o)=(BC/2)/(√(3)-1) ----> by CAH (adjacent side divided buy the hypotenuse

Remember that


cos(30^o)=(√(3))/(2)

substitute


(√(3))/(2)=(BC/2)/(√(3)-1)


BC=(3-√(3))\ units

Find the perimeter


P=3(√(3)-1)+3(3-√(3))


P=3√(3)-3+9-3√(3))


P=6\ units

a square is constructed on each side of an equilateral triangle, and segments are-example-1
User Toby Mellor
by
7.0k points
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