SOLUTION

If a square region has area n, what is the length of the diagonal of the square in terms of n ?

(A) \(\sqrt{2n}\)
(B) \(\sqrt{n}\)
(C) \(2\sqrt{n}\)
(D) 2n
(E) 2n^2

The area of a square is \(\frac{diagonal^2}{2}\).

\(\frac{diagonal^2}{2}=n\) --> \(diagonal=\sqrt{2n}\).

Answer: A.
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Area of square = n,
Hence each of the sides of the square are \(\sqrt{n}\)

Now the sides of any rectangular isosceles triangle have a ratio of 1:1:\(\sqrt{2}\).

Applying this ratio to a rectangular isosceles triangle whose equal sides are \(\sqrt{n}\), the ratio of the sides for this triangle becomes:
\(\sqrt{n}:\sqrt{n}:\sqrt{2}\sqrt{n}\)

Which is equivalent to
\(\sqrt{n} : \sqrt{n} : \sqrt{2n}\)

Answer: A

Ans A

Area of square=n=(side)^2
\sqrt{n}=side
(Diagonal)^2= (\sqrt{n})^2 + (\sqrt{n})^2
Diagonal=\sqrt{n+n}
=\sqrt{2n}

Hi All,

This question can be solved by TESTing VALUES.

We're told that a square has an AREA of N. We're asked for the length of the diagonal of that square.....

IF...
Side of the square = 3....
the area of the square = N = (3)(3) = 9....
the diagonal of the square = 3√2

Answer A: √(2N) = √(18) = 3√2 This IS a match.
Answer B: √N = √9 = 3 NOT a match.
Answer C: 2√N = 2√9 = 6 NOT a match.
Answer D: 2N = 18 NOT a match.
Answer E: 2N^2 = 162 NOT a match.

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Area = s^2
Area represented as n
n = s^2
each side s = \(\sqrt{n}\)
Diagonal of a square =\(s\sqrt{2}\)
Diagonal = \(\sqrt{n}\)\(\sqrt{2}\)
\(\sqrt{2n}\)
A

Hi Bunuel,

can you please explain one thing. Square has all sides of equal lenght. And if Diagonal divides square into two triangles.

How can such triangle be a RIGHT trangle ? It has two sides equal and two angles equal which is an Isosceles Triangle. Right?

Have an awesome day

Don't we have a right angle there? We'll get two isosceles right triangles. You could draw a square and divide it into two triangles with a diagonal, should be easy to check.
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Hi dave13,

When you cut a square in half (from corner to opposite corner), you always end up with two 45/45/90 right triangles (and yes, they are Isosceles triangles). With tougher Geometry questions on the GMAT, you should look for common shapes (such as right triangles) inside of larger (or more complex) shapes. Sometimes the 'key' to solving those types of prompts is to redraw a picture by 'breaking' it into pieces and then using the appropriate Geometry formulas to complete the calculations.

GMAT assassins aren't born, they're made,
Rich
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let side of square is \(x\)
so, \(area=x^2\)
\(x^2=n\)
\(x=\sqrt{n}\)
As the diagonal length of a square with side length \(x\) is = \(\sqrt{2}x\)
so, diagonal length of square with side length \(\sqrt{n}\) = \(\sqrt{2}\sqrt{n}\) = \(\sqrt{2n}\)

Answer A

Alternately you can use pathagoras theorem to find diagonal length of square with side length \(\sqrt{n}\)

Area of square = x^2

Since we are given the area = n then the side equals \(\sqrt{n}\)

A diagonal in a square creates two 45:45:90 triangles.

45:45:90
x:x:x\(\sqrt{2}\)
since the side equals \(\sqrt{n}\) then the diagonal would be \(\sqrt{n} * \sqrt{2} = \sqrt{2n}\)

Answer choice A

Area = n = a^2 ; where a is the side of the square

\(a = \sqrt{n}\)

Diagonal = \(a\sqrt{2} = \sqrt{n}\sqrt{2} = \sqrt{2n}\)

IMO A
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\(A = \frac{d^2}{2}\)

Or, \(n = \frac{d^2}{2}\)

Or, \(2n = d^2\)

So, \(d = \sqrt{2n}\), Answer must be (A)
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