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655-705 (Hard)|   Geometry|      
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Bunuel
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The figure above shows a square inscribed within an equilateral triang 23 Dec 2014, 09:37
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Tough and Tricky questions: Word Problems.




The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36
B. 36√3
C. 24√3 + 24
D. 24√3 + 36
E. 72


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D
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PareshGmat
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The figure above shows a square inscribed within an equilateral triang 24 Dec 2014, 01:32
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Answer = D. 24√3 + 36

Refer diagram below:

Attachment:
square_inscribed_in_triangle.gif
square_inscribed_in_triangle.gif [ 5.8 KiB | Viewed 29504 times ]

\(\triangle\) ABC is similar to \(\triangle\) ADC

\(\frac{AD}{AB} = \frac{DE}{BC}\)

\(\frac{AD}{AB} = \frac{12}{AB}\)(Because AB = BC in equilateral triangle)

AD = 12

So, AD = AE = DE = 12 (Equilateral triangle)

Let BG = FC = a, then BD = 2a

\((2a)^2 = a^2 + 12^2\)

\(3a^2 = 12*12\)

\(a^2 = 16*3\)

\(a = 4\sqrt{3}\)

\(Perimeter = 6(a+6) = 6(4\sqrt{3} + 6) = 24\sqrt{3} + 36\)
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The figure above shows a square inscribed within an equilateral triang 23 Dec 2014, 10:00
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Solution: Refer to attached figure:
Triangle BDG is 30-60-90 ==> BD = 2x / √3 = 8√3
& BG = x / √3 = 4√3
Similarly, Triangle CFE is 30-60-90 ==> CE = 2x / √3 = 8√3
& FC = x / √3 = 4√3

Triangle ADE is Equlateral ==> AD = AE = x = 12

Perimeter = AD + BD + AE + CE + BG + GF + FC = 24√3 + 36

Answer D
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The figure above shows a square inscribed within an equilateral triang 08 Jan 2015, 07:34
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Bunuel

Tough and Tricky questions: Word Problems.




The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36
B. 36√3
C. 24√3 + 24
D. 24√3 + 36
E. 72


Kudos for a correct solution.
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Attachment:
square_inscribed_in_triangle.gif
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OFFICIAL SOLUTION:

D) If you look at small triangles to the left and to the right of the square, you’ll notice that they are 90°-60°-30° triangles. Furthermore, the triangles have equal sides, so these triangles are equal.

The sides of a 90°-60°-30° triangle are in 2:√3:1 ratio. If we denote the smallest side in this triangle by y, then x : y = √3 : 1. Therefore y = 12 / √3.Let’s simplify to exclude a root sign in the denominator. 12 / √3 = (12 × √3) / (√3 × √3) = 4√3.

The base side of the large equilateral triangle equals to the sum of the three segments. y + x + y = 4√3 + 12 + 4√3 = 12 + 8√3.

All sides of an equilateral triangle are equal. Therefore the perimeter of the triangle is 3 × (12 + 8√3) = 36 + 24√3. The correct answer is D.

Alternative solution:

Is there a way to solve the problem if I don’t remember the properties of a 90°-60°-30° triangle?

Yes, there is. The small triangle on top of the square is equilateral as well, because all its angles are 60°. Therefore all its sides are equall x (12 inches).

The hypotenyses of the right triangles to the left and to the right of the square are both equal √(x² + y²) according to Pythagorean theorem.

Therefore the two sides of the large equilateral triangle equal x + √(x² + y²). The base side of the large equilateral triangle equals x + 2y. All sides in an equilateral triangle must be equal. Therefore x + √(x² + y²) = x + 2y.

x = √3y (because we deal with positive values only)
y = 4√3 so we can easily calculate the perimeter.
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The figure above shows a square inscribed within an equilateral triang 05 Aug 2017, 21:49
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Bunuel

Tough and Tricky questions: Word Problems.




The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36
B. 36√3
C. 24√3 + 24
D. 24√3 + 36
E. 72


Kudos for a correct solution.
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Attachment:
square_inscribed_in_triangle.gif
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Excellent problem.

First derived that the square had made an equilateral triangle above it and then used the 30-60-90 ratio to find the length of the remaining portion.
Hence the actual perimeter is 3*(12+8\sqrt{3}) = 36+24\sqrt{3}

Option D is the answer.......
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The figure above shows a square inscribed within an equilateral triang 07 Aug 2017, 09:07
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Bunuel
Bunuel

Tough and Tricky questions: Word Problems.




The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36
B. 36√3
C. 24√3 + 24
D. 24√3 + 36
E. 72


Kudos for a correct solution.
Show Spoiler
Attachment:
square_inscribed_in_triangle.gif

OFFICIAL SOLUTION:

D) If you look at small triangles to the left and to the right of the square, you’ll notice that they are 90°-60°-30° triangles. Furthermore, the triangles have equal sides, so these triangles are equal.

The sides of a 90°-60°-30° triangle are in 2:√3:1 ratio. If we denote the smallest side in this triangle by y, then x : y = √3 : 1. Therefore y = 12 / √3.Let’s simplify to exclude a root sign in the denominator. 12 / √3 = (12 × √3) / (√3 × √3) = 4√3.

The base side of the large equilateral triangle equals to the sum of the three segments. y + x + y = 4√3 + 12 + 4√3 = 12 + 8√3.

All sides of an equilateral triangle are equal. Therefore the perimeter of the triangle is 3 × (12 + 8√3) = 36 + 24√3. The correct answer is D.

Alternative solution:

Is there a way to solve the problem if I don’t remember the properties of a 90°-60°-30° triangle?

Yes, there is. The small triangle on top of the square is equilateral as well, because all its angles are 60°. Therefore all its sides are equall x (12 inches).

The hypotenyses of the right triangles to the left and to the right of the square are both equal √(x² + y²) according to Pythagorean theorem.

Therefore the two sides of the large equilateral triangle equal x + √(x² + y²). The base side of the large equilateral triangle equals x + 2y. All sides in an equilateral triangle must be equal. Therefore x + √(x² + y²) = x + 2y.

x = √3y (because we deal with positive values only)
y = 4√3 so we can easily calculate the perimeter.
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Dear Bunuel,
Hi. I really don't get how you guys get the small triangle is 30-60-90?
Would you explain? Thank you
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The figure above shows a square inscribed within an equilateral triang 07 Aug 2017, 09:59
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Bunuel
Bunuel

Tough and Tricky questions: Word Problems.




The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36
B. 36√3
C. 24√3 + 24
D. 24√3 + 36
E. 72


Kudos for a correct solution.
Show Spoiler
Attachment:
square_inscribed_in_triangle.gif

OFFICIAL SOLUTION:

D) If you look at small triangles to the left and to the right of the square, you’ll notice that they are 90°-60°-30° triangles. Furthermore, the triangles have equal sides, so these triangles are equal.

The sides of a 90°-60°-30° triangle are in 2:√3:1 ratio. If we denote the smallest side in this triangle by y, then x : y = √3 : 1. Therefore y = 12 / √3.Let’s simplify to exclude a root sign in the denominator. 12 / √3 = (12 × √3) / (√3 × √3) = 4√3.

The base side of the large equilateral triangle equals to the sum of the three segments. y + x + y = 4√3 + 12 + 4√3 = 12 + 8√3.

All sides of an equilateral triangle are equal. Therefore the perimeter of the triangle is 3 × (12 + 8√3) = 36 + 24√3. The correct answer is D.

Alternative solution:

Is there a way to solve the problem if I don’t remember the properties of a 90°-60°-30° triangle?

Yes, there is. The small triangle on top of the square is equilateral as well, because all its angles are 60°. Therefore all its sides are equall x (12 inches).

The hypotenyses of the right triangles to the left and to the right of the square are both equal √(x² + y²) according to Pythagorean theorem.

Therefore the two sides of the large equilateral triangle equal x + √(x² + y²). The base side of the large equilateral triangle equals x + 2y. All sides in an equilateral triangle must be equal. Therefore x + √(x² + y²) = x + 2y.

x = √3y (because we deal with positive values only)
y = 4√3 so we can easily calculate the perimeter.

Dear Bunuel,
Hi. I really don't get how you guys get the small triangle is 30-60-90?
Would you explain? Thank you
Show more

Every angle of an equilateral triangle is 60 degrees so the rightmost angle of the small triangle is 60 (since it is also the angle of the large equilateral triangle). There is also a 90 degree angle made by the side of the square. So the leftover angle must be 30 degrees. Hence the 30-60-90 triangle.
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The figure above shows a square inscribed within an equilateral triang 07 Aug 2017, 11:09
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pclawong
Bunuel
Bunuel

Tough and Tricky questions: Word Problems.




The figure above shows a square inscribed within an equilateral triangle where the top of the square is parallel to the bottom of the triangle. If the side x of the square measures 12 inches, then what is the perimeter of the equilateral triangle (in inches)?

A. 36
B. 36√3
C. 24√3 + 24
D. 24√3 + 36
E. 72


Kudos for a correct solution.
Show Spoiler
Attachment:
square_inscribed_in_triangle.gif

OFFICIAL SOLUTION:

D) If you look at small triangles to the left and to the right of the square, you’ll notice that they are 90°-60°-30° triangles. Furthermore, the triangles have equal sides, so these triangles are equal.

The sides of a 90°-60°-30° triangle are in 2:√3:1 ratio. If we denote the smallest side in this triangle by y, then x : y = √3 : 1. Therefore y = 12 / √3.Let’s simplify to exclude a root sign in the denominator. 12 / √3 = (12 × √3) / (√3 × √3) = 4√3.

The base side of the large equilateral triangle equals to the sum of the three segments. y + x + y = 4√3 + 12 + 4√3 = 12 + 8√3.

All sides of an equilateral triangle are equal. Therefore the perimeter of the triangle is 3 × (12 + 8√3) = 36 + 24√3. The correct answer is D.

Alternative solution:

Is there a way to solve the problem if I don’t remember the properties of a 90°-60°-30° triangle?

Yes, there is. The small triangle on top of the square is equilateral as well, because all its angles are 60°. Therefore all its sides are equall x (12 inches).

The hypotenyses of the right triangles to the left and to the right of the square are both equal √(x² + y²) according to Pythagorean theorem.

Therefore the two sides of the large equilateral triangle equal x + √(x² + y²). The base side of the large equilateral triangle equals x + 2y. All sides in an equilateral triangle must be equal. Therefore x + √(x² + y²) = x + 2y.

x = √3y (because we deal with positive values only)
y = 4√3 so we can easily calculate the perimeter.

Dear Bunuel,
Hi. I really don't get how you guys get the small triangle is 30-60-90?
Would you explain? Thank you
Show more

Hello!

Each angle of a square is 90 degrees; so the adjacent angle [one of the angle in small triangle] is 180-90 = 90 degrees.
Every angle of an equilateral triangle is 60 degrees and we have the same angle common in the small triangle - hence it is 60 .
So the third angle is 180-(60=90) = 30 degrees.

Hence it is 30 60 90 triangle and the sides are in the ratio 1: sqrt3 :2 and we can proceed further.
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The figure above shows a square inscribed within an equilateral triang 28 Jul 2021, 07:55
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beattheheat
Solution: Refer to attached figure:
Triangle BDG is 30-60-90 ==> BD = 2x / √3 = 8√3
& BG = x / √3 = 4√3
Similarly, Triangle CFE is 30-60-90 ==> CE = 2x / √3 = 8√3
& FC = x / √3 = 4√3

Triangle ADE is Equlateral ==> AD = AE = x = 12

Perimeter = AD + BD + AE + CE + BG + GF + FC = 24√3 + 36

Answer D
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How did you get 8√3?
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