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# An equilateral triangle and three squares are combined as shown above,

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Math Expert
Joined: 02 Sep 2009
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An equilateral triangle and three squares are combined as shown above,  [#permalink]

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20 Sep 2016, 07:41
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Difficulty:

45% (medium)

Question Stats:

75% (02:10) correct 25% (02:23) wrong based on 139 sessions

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An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares?

A. 18
B. 27
C. 36
D. 48
E. 64

Attachment:

T6027.png [ 5.85 KiB | Viewed 2923 times ]

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Re: An equilateral triangle and three squares are combined as shown above,  [#permalink]

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20 Sep 2016, 09:16
triangle area = root(3)s^2/4
area of 3 squares together = 3 s^2

root(3)s^2/4 + 3 s^2 = 48 +4root(3)
root(3)/4s^2 = 4root(3)

s^2 = 16 => s = 4

there are 3 sides of each square = 3(4)(3) = 36

Option C
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An equilateral triangle and three squares are combined as shown above,  [#permalink]

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20 Sep 2016, 12:39
Bunuel wrote:

An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares?

A. 18
B. 27
C. 36
D. 48
E. 64

Attachment:
T6027.png

We are given three square and equilateral triangle areas combines is 48+4√3.

Now we can write as $$3a^2$$ + √3/4( $$a^2$$ ) = 48+4√3.

=> $$a^2$$ ( 3 + √3/4) = 4(12+√3)

=> $$a^2$$ ( (12+√3/4) ) = 4(12+√3)
=> $$a^2$$ (1/4) = 4
=> $$a^2$$ = 16
a = 4.

We need to take only external sides of the figures , overall we have 9 sides of equal length , we get 9*4 = 36

IMO option C.
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Re: An equilateral triangle and three squares are combined as shown above,  [#permalink]

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22 Oct 2016, 23:34
The Q is formula based

Total area of square and triangle = ( 3+sq rt3/4) * Side^2

( 3+sq rt3/4) * Side^2 = 48 +4 sq rt3

=> Side^2 = 16
=>Side = 4

Perimeter is sum of external sides = 9 * 4 = 36 C
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Re: An equilateral triangle and three squares are combined as shown above,  [#permalink]

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23 Oct 2016, 00:45
1
Bunuel wrote:

An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares?

A. 18
B. 27
C. 36
D. 48
E. 64

Attachment:
The attachment T6027.png is no longer available

Attachment:

T6027.png [ 7.17 KiB | Viewed 2019 times ]

There are 4 shapes -

1. 3 Squares of sides a
2. 1 Triangle of side a

Area of 3 Squares = $$3*a^2$$

Area of Triangle = $$\frac{√3}{4}a^2$$

Now, $$\frac{√3}{4}a^2$$ + $$3a^2$$ = $$48+4√3$$

Or, $$a^2(\frac{√3}{4}$$ + $$3 )$$ = $$48+4√3$$

Or, $$a^2( √3$$ + $$12)$$ = $$4 ( 48+4√3 )$$

Or, $$a^2( √3$$ + $$12)$$ = $$192 + 16√3$$

Or, $$a^2 = 16$$

Or, $$a = 4$$

Quote:
What is the perimeter of the shape formed by the triangle and squares?

Count the number of sides in the figure = 9

So, Perimeter is $$9*4 = 36$$

Hence answer will be (C)
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Re: An equilateral triangle and three squares are combined as shown above,  [#permalink]

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25 Nov 2017, 08:23
1
Bunuel wrote:

An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares?

A. 18
B. 27
C. 36
D. 48
E. 64

Attachment:
The attachment T6027.png is no longer available

Assume the second term, $$4\sqrt{3}$$, of the given area is a clue: it's likely to be the area of the equilateral triangle.
Divide 48 by 3 (squares) = 16. Square side = 4.
Check: Length = 4 must equal equilateral triangle side. Area of equilateral triangle if side = 4 is $$\frac{4^2\sqrt{3}}{4} = 4\sqrt{3}$$ - Correct
Square side = 4, there are nine sides on the perimeter, 9 * 4 = 36

Longer version

Area generally: formulas and given value

Knowing the formula for the area of an equilateral triangle makes this problem a lot easier.*

Total area = (area of three squares) + (area of equilateral triangle)

Area of three square $$(3)s^2$$

Area of equilateral triangle = $$\frac{s^2\sqrt{3}}{4}$$

Total area: $$3s^2 + \frac{s^2\sqrt{3}}{4}$$

Total area = $$48 + 4√3$$

Find side length of square from area of square set equal to 48

Gamble a little. The second given term, $$4\sqrt{3}$$ is likely to be the area of the equilateral triangle.
So assume that the integer portion of the given area will yield a square's side length

Set just the integer portion of the area, 48, equal to the area of the three squares

$$3s^2 = 48$$
$$s^2 = 16$$
$$s = 4$$

Check: square side = triangle side, find triangle area

Each square: has equal side lengths that are equal to the side of the equilateral triangle (each square shares a side with the triangle)
If square side length = 4, triangle side length must = 4

Set the other given term equal to the area of the equilateral triangle
$$\frac{s^2\sqrt{3}}{4} =4√3$$

$${s^2\sqrt{3}}=16√3$$

$$s^2 = 16$$
$$s = 4$$

Correct: the side of square = side of triangle = 4

Perimeter

Perimeter = 9 square side lengths: $$9 * 4 = 36$$

*If you don't remember the formula for the area of an equilateral triangle, draw one. Drop an altitude, which is a perpendicular bisector of the opposite side and of the vertex.
That altitude creates two congruent right 30-60-90 triangles with side lengths that correspond to 30-60-90, in ratio $$x : x\sqrt{3} : 2x$$

Side lengths? Side opposite the 90° angle = $$2x = 4$$. Side opposite 30° angle is half of that, i.e., $$x, x = 2$$. Side opposite 60° angle = height of triangle = $$x\sqrt{3}$$ or $$2\sqrt{3}$$

Area$$=\frac{b*h}{2} = (4*2\sqrt{3})*\frac{1}{2} = 4\sqrt{3}$$. Correct: it is equal to the second term in the area given.

Attachment:

equitriarea.png [ 40.47 KiB | Viewed 1311 times ]
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An equilateral triangle and three squares are combined as shown above,  [#permalink]

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28 Nov 2017, 00:01
1
genxer123 wrote:
Bunuel wrote:

An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares?

A. 18
B. 27
C. 36
D. 48
E. 64

Attachment:
T6027.png

Assume the second term, $$4\sqrt{3}$$, of the given area is a clue: it's likely to be the area of the equilateral triangle.
Divide 48 by 3 (squares) = 16. Square side = 4.
Check: Length = 4 must equal equilateral triangle side. Area of equilateral triangle if side = 4 is $$\frac{4^2\sqrt{3}}{4} = 4\sqrt{3}$$ - Correct
Square side = 4, there are nine sides on the perimeter, 9 * 4 = 36

The quick answer is so brilliant genxer123

Thanks
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Re: An equilateral triangle and three squares are combined as shown above,  [#permalink]

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28 Nov 2017, 02:02
Perimeter is 9a.

So it's a multiple of 9.
Eliminate D, E

Now, take the hint from area (root part) and find the side of square.

So side comes out to be 4

9*4=36

C

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Re: An equilateral triangle and three squares are combined as shown above,  [#permalink]

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29 Nov 2017, 17:14
Bunuel wrote:

An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares?

A. 18
B. 27
C. 36
D. 48
E. 64

Attachment:
T6027.png

We see that the squares and the triangle have equal sides. If we let the side of either shape = n, the total area is:

(n^2√3)/4 + 3n^2 = 48+4√3

n^2√3 + 12n^2 = 192 + 16√3

Looking at the equation we see that:

n^2√3 = 16√3

And

12n^2 = 192

In either equation, we have n = 4. Thus, the perimeter is 4 x 9 = 36.

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Re: An equilateral triangle and three squares are combined as shown above, &nbs [#permalink] 29 Nov 2017, 17:14
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