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What is the perimeter of the triangle?

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by himu » Thu Apr 04, 2013 8:47 pm
A square with area 16 is perfectly inscribed inside an equilateral triangle. What is the perimeter of the triangle?
(83√)3+4
43√+4
83√+12
24+12
(323√)3+12
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by Anju@Gurome » Thu Apr 04, 2013 9:04 pm
himu wrote:A square with area 16 is perfectly inscribed inside an equilateral triangle. What is the perimeter of the triangle?
Refer to the diagram below,
Image
Area of the square PQRS is 16.
So, PQ = QR = RS = SP = 4

Triangle APQ is a 30-60-90 triangle with PQ = 4.
Hence, AQ = 4/√3

Similarly, RB = 4/√3

So, AB = AQ + QR + RB = 4/√3 + 4 + 4/√3 = 4 + 8/√3
--> Perimeter of the triangle = 3*(4 + 8/√3) = (12 + 8√3)

Your options are not properly typed.
But it seems C is the correct answer.
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by srcc25anu » Thu Apr 04, 2013 9:11 pm
Tr ABC is equilateral hence Angles A, B and C are 60 degrees
DEGF is a square with side 4 each

In Tr BDF: DF = 4
following traingle property 30:60:90 sides are in the ratio 1:root3:2
hence we know side corresponding to root3 = 4
therefore side corr. to 2 or 90 degrees = 8/root3 = BD

in Tr ADY: DY = half of side of square or 2
again forllowing the 1:root3:2 property we know 1 corresponds to 2
thus 2 would correspond to 4 which is = AD

Side of Tr ABC = 4 + 8/root3
Perimeter = 3 * side => 12 + 24/root3
multiply both numerator and denomintor by root3
P = 12 + 24 * root3 / 3 or 12 + 8 * root3
Hence C is answer
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