The Side Of An Equilateral Triangle Has The Same Length As The Diagonal GMAT Data Sufficiency

Question: 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 √3.
(2) The area of the equilateral triangle is equal to 36 √6.

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: (D)

Solution and Explanation:

Approach Solution 1:

Remember these characteristics: [a] A diagonal divides a square into triangles with sides ranging from 1:1:2 to 45:45:90.
[b] A square's area equals s^2, where s is the square's side.
[c] Height divides the triangle into two equal triangles of sides 1:3:2, 30 60, and 90.
[d] The area of an equilateral triangle is equal to ½ * t*h, where t is a triangle's side and h is its height. and [c], which reduces to (3t^2) /4

Since you can solve this question without using any computations, the attributes are crucial. Your understanding of properties will be tested by this query!
(1) The equilateral triangle measures 63 in height.
The sides [c] and the area [d] may be found if we know the height! We can also find the sides [a] and the area [b] if we know the side and the square's diagonal (which is shown in the question stem). Sufficient.

(2) The equilateral triangle has a 363 surface area.
If we are familiar with the area, we can locate the sides [d], then read (1). Sufficient.

Approach Solution 2:

1) Two right triangles with angles 30, 60, and 90, and ratios of sides 1, 3, 2 are equal to an equilateral triangle divided by height.
Hypotenuse = 6*2 = 12 if height = 63. This hypotenuse is the diagonal of a square and a side of an equilateral triangle.
Square area equivalent to d^2 / 2 from diagonal: 12^2 / 2=72
Sufficient

2) Equilateral triangle's area is equal to s^2 sqrt(3) / 4, where s is a triangle side.
Given that we are familiar with the area 363363, we can convert it to a formula by multiplying it by 4: 436sqrt(3)4436sqrt(3)
Therefore, s^2=144 --> s=12 and this is the square's diagonal.
Square area equivalent to d^2/2 from diagonal: 12^2 / 2 = 72
Sufficient

Approach Solution 3:

All sides,area and other line segments such as altitude, median, diagonal etc are interconnected in both equilateral triangle and square.

Therefore, if we know any one of these measurements, it will help us to derive area,circumferece etc of each equilateral triangle and square.
The question states the length of the side of the equilateral triangle is equal to another line segment i.e diagonal of square ..
Therefore, only by knowing one measurement of square or triangle, we can answer the problem.

1. The height of the equilateral triangle is equal to 6 √3.
   Since height is given.... Therefore, it is sufficient.
2. The area of the equilateral triangle is equal to 36 √6.
   Since area is given... Therefore, it is sufficient.

“The side of an equilateral triangle has the same length as the diagonal of a square”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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