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# The side of an equilateral triangle has the same length as the diagona

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The side of an equilateral triangle has the same length as the diagona  [#permalink]

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23 Jun 2015, 23:11
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The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

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Re: The side of an equilateral triangle has the same length as the diagona  [#permalink]

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24 Jun 2015, 04:02
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Quote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

Properties to remember:
[a] Diagonal divides the square to 2 equal 45 45 90 triangles with sides ratios 1:1:$$\sqrt{2}$$
[b] Area of square = $$s^2$$, where s is the side of the square.

[c] Height divides the triangle into 2 equal 30 60 90 triangles with side ratios 1:$$\sqrt{3}$$:2
[d] Area of equilateral triangle = $$\frac{1}{2}t*h$$, where t is the side of the triangle and h is the height. and using [c] that simplifies to $$\sqrt{3}t^2/4$$

The properties for this question are important, because you don't need to make any calculations to solve it! This question is to test your knowledge of properties!

(1) The height of the equilateral triangle is equal to 63‾√.

If we know the height we can find: sides [c] and the area [d]! And if we know the side, we know the diagonal of the square (given in question stem), and knowing that we can find sides [a] and the area [b]. Sufficient.

(2) The area of the equilateral triangle is equal to 363‾√.

If we know the area, we can find the sides [d], from there read (1). Sufficient.

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The side of an equilateral triangle has the same length as the diagona  [#permalink]

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24 Jun 2015, 04:28
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Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

1) equilateral traingle divided by height equal to two right triangles with angles 30, 60, 90 and ratios of sides $$1, \sqrt{3}, 2$$
So if height = $$6\sqrt{3}$$ then hypotenuse = 6*2 = 12. This hypotenuse is side of equilateral triangle and diagonal of square.
Area of square from diagonal equal to $$\frac{d^2}{2}$$ --> $$\frac{12^2}{2} = 72$$
Sufficient

2) area of equilateral triangle equal to $$\frac{s^2*sqrt(3)}{4}$$ where s is side of the triangle
As we know area $$36\sqrt{3}$$ we can transform it to formula by multipling on 4 --> $$\frac{4* 36*sqrt(3)}{4}$$
So $$s^2 = 144$$ --> $$s = 12$$ and this is diagonal of square
Area of square from diagonal equal to $$\frac{d^2}{2}$$ --> $$\frac{12^2}{2} = 72$$
Sufficient

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The side of an equilateral triangle has the same length as the diagona  [#permalink]

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24 Jun 2015, 06:14
Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

The basic methods and Properties have already been mentioned so let me use more of DS technique to answer this question

Question : What is the area of the square?

To answer the question we only need to know the Dimension of side or Diagonal of Square as the two, in Square, can be related by $$Diagonal = Side*\sqrt{2}$$

Question : What is the Diagonal or Side of Square?

Given: The side of an equilateral triangle has the same length as the diagonal of a square
Since the diagonal of Square is related with the the Side of Equilateral triangle so now any information about any dimension (Side of Height or Area or Perimeter) of equilateral triangle will get us the answer.

Area of Equilateral Triangle = $$[\sqrt{3}/4]*Side^2$$
Perimeter of Equilateral Triangle = $$3*Side$$
Height of Equilateral Triangle = $$[\sqrt{3}/2]*Side$$

Question : What is Side of Height or Area or Perimeter of Equilateral Triangle?

Statement 1:The height of the equilateral triangle is equal to $$6\sqrt{3}$$

SUFFICIENT

Statement 2:The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

SUFFICIENT

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The side of an equilateral triangle has the same length as the diagona  [#permalink]

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24 Jun 2015, 10:39
Area of square= side square. Side of square can be found out if we know height of the triangle (by applying Pythagorean theorem, and then putting the same value as diagonal of square )
A) height of triangle =6roo3 , so area can be found out so sufficient
B) area of triangle is 36root 3, so area of square can be found out, sufficient

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Re: The side of an equilateral triangle has the same length as the diagona  [#permalink]

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24 Jun 2015, 22:27
1
Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

Note that squares and equilateral triangles are perfectly symmetrical figures. If you have any one dimension for them (side/altitude/diagonal/area), you can get everything else. You are given the relation between the side of the triangle and the diagonal of the square. This means that if you get any one dimension for any one figure, you will be able to calculate everything else for both the figures.
Each statement gives you one dimension and hence each statement alone will be sufficient to get the area of the square.

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Re: The side of an equilateral triangle has the same length as the diagona  [#permalink]

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24 Jun 2015, 22:50
Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

All sides,area and other line segments such as median,altitude,diaginal median etc are interrelated in both square and equilateral triangle..
So knowing any one of these can help us in finding area,circumferece etc of each...
In this question one of the side is related to another line segment of square ..
therefore , by just knowing even one measurement of square or triangle is enough..
1) height given.... suff
2) area given... suff
ans D
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Re: The side of an equilateral triangle has the same length as the diagona  [#permalink]

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29 Jun 2015, 07:13
Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

No calculation is needed to solve this problem. Both equilateral triangles and squares are regular figures— those that can change size, but never shape.

Regular figures (squares, equilaterals, circles, spheres, cubes, 45-45-90 triangles, 30-60-90 triangles, and others) are those for which you only need one measurement to know every measurement. For instance, if you have the radius of a circle, you can get the diameter, circumference, and area. If you have a 45-45-90 or 30-60-90 triangle, you only need one side to get all three. In this problem, if you have the side of an equilateral, you could get the height, area, and perimeter. If you have the side of a square, you could get the diagonal, area, and perimeter.

If you have two regular figures, as you do in this problem, and you know how they are related numerically (“the side of an equilateral triangle has the same length as the diagonal of a square”), then you can safely conclude that any measurement for either figure will give you any measurement for either figure.

The question can be rephrased as, “What is the length of any part of either figure?”

1) This gives you the height of the triangle. SUFFICIENT.
2) This gives you the area of the triangle. SUFFICIENT.

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Re: The side of an equilateral triangle has the same length as the diagona  [#permalink]

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12 Apr 2016, 14:33
from statement 1 we get height of equilateral triangle
Height of triangle ----> Side of triangle ----> Diagonal of square ----> Side of square ----> Area of square
statement 1 sufficient
From statement 2 we get Area of equilateral triangle
Area of triangle ----> Side of triangle ----> Diagonal of square ----> Side of square ----> Area of square
statement 2 also sufficient
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Re: The side of an equilateral triangle has the same length as the diagona  [#permalink]

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05 Sep 2017, 16:46
Bunuel wrote:
The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

(1) The height of the equilateral triangle is equal to $$6\sqrt{3}$$.
(2) The area of the equilateral triangle is equal to $$36\sqrt{3}$$.

Kudos for a correct solution.

Statement 1-

If we know the height of the equilateral triangle then we can just work back words to find the side lengths of the triangle

Suff

Statement 2

If we know the area of the equilateral triangle then we just could apply the alternative formula for the area of an equilateral triangle

\sqrt{3}/4 = area of equilateral

S^2 \sqrt{3}/4 = $$6\sqrt{3}$$

S^2 \sqrt{3} =144 \sqrt{3}

Suff

D
Re: The side of an equilateral triangle has the same length as the diagona   [#permalink] 05 Sep 2017, 16:46
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# The side of an equilateral triangle has the same length as the diagona

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