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655-705 (Hard)|   Geometry|      
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Bunuel
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The figure shows a square surrounded by an equilateral triangle. If 24 Jan 2022, 08:41
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D
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Difficulty:

65% (hard)

Question Stats:

62% (02:44) correct 38% (02:23) wrong based on 76 sessions

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The figure shows a square surrounded by an equilateral triangle. If the edge x of the inscribed square is 12 inches long, then what is the perimeter of the equilateral triangle ?

A. \(36\) inches

B. \(36\sqrt{3}\) inches

C. \(36\sqrt{3}+24\) inches

D. \(24\sqrt{3}+36\) inches

E. \(72\) inches


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D
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The figure shows a square surrounded by an equilateral triangle. If 24 Jan 2022, 17:15
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Bunuel

The figure shows a square surrounded by an equilateral triangle. If the edge x of the inscribed square is 12 inches long, then what is the perimeter of the equilateral triangle ?

A. \(36\) inches

B. \(36\sqrt{3}\) inches

C. \(36\sqrt{3}+24\) inches

D. \(24\sqrt{3}+36\) inches

E. \(72\) inches

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Breaking Down the Graph:

The top shaded triangle is an equilateral triangle with a side of 12 inches. The other two smaller shaded triangles can be combined the build an equilateral triangle with a height of 12 inches, which will has a side length of \(12*\frac{2}{\sqrt{3}} = 8\sqrt{3}\)

Then the perimeter of the entire triangle is 12 + 12 from the top triangle, \(8\sqrt{3}*3\) from the left and right shaded triangles, and 12 from the bottom of the square. Thus \(36 + 24\sqrt{3}\) in total.

Answer: D
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JyothisJacob
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The figure shows a square surrounded by an equilateral triangle. If 14 Feb 2022, 07:49
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2 concepts used:
1. Triangle above is equilateral with side = 12
2. 30-60-90 Triangle

Using both compute the length of side x 3 = perimeter
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The figure shows a square surrounded by an equilateral triangle. If 14 Feb 2022, 23:46
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The upper shaded triangle is a 60, 60, 60 triangle
as the side of the square is parallel to the base of the equilateral triangle.

the remaining two triangles are 30, 60, 90 triangles
with side opposite to 60 degree = 12
using the 30 60 90 formula = x, root 3, 2


we can figure out the remaining side of both small triangles, 2(4 root 3 + 8 root 3 )
+12 = base of the square on the base of the triangle
when we add it all
answer is D

with base = 12 so all sides = 12.
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The figure shows a square surrounded by an equilateral triangle. If 26 Feb 2023, 07:11
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Given: The figure shows a square surrounded by an equilateral triangle.
Asked: If the edge x of the inscribed square is 12 inches long, then what is the perimeter of the equilateral triangle ?

x = 12
Side of the equilateral triangle = \(12 + 12*2/\sqrt{3}\)
Parameter of the equilateral triangle = \(3*(12 + 12*2/\sqrt{3}) = 36 + 24\sqrt{3}\)

IMO D
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The figure shows a square surrounded by an equilateral triangle. If 04 Mar 2025, 07:17
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