Problem Solving

Hello,

For the following:

Points A and C have (x,y) coordinates (-1,3) and (2,-1) respectively, and are
opposite vertices of square ABCD, what is the perimeter of square ABCD?


OA: [spoiler]10 sq. root (2)[/spoiler]


I was a bit confused by this question since it doesn't appear to be a square as shown in the diagram:

Thanks a lot,
Sri

Sri,

You assumed that the sides of the squares are parallel to the x and y axes. As you drawing shows, this is not possible since the shape you drew is not a square. Think of the question in more simple terms:

Given the length of a diagonal (the distance between two opposite vertices), what is the perimeter of the square? If you find the length of AC, you will have the first piece!


-Patrick

gmattesttaker2 wrote: Points A and C have (x,y) coordinates (-1,3) and (2,-1) respectively, and are
opposite vertices of square ABCD, what is the perimeter of square ABCD?

When no figure is given, there is a likely reason:
If a figure were given, the problem would be too EASY.
Implication:
Be careful when drawing.
The problem probably contains a TWIST.


As the figure above shows, AC is the hypotenuse of a 3-4-5 triangle.
Thus, AC = 5.

AC is also the diagonal of square ABCD.
The diagonal of a square = s√2.
Since AC = 5, we get:
s√2 = 5.
s = 5/√2.

Thus:
p = 4s = 4 * (5/√2) = (4*5*√2) / (√2*√2) = (20√2)/2 = [spoiler]10√2[/spoiler].

Patrick_GMATFix wrote:Sri,

You assumed that the sides of the squares are parallel to the x and y axes. As you drawing shows, this is not possible since the shape you drew is not a square. Think of the question in more simple terms:

Given the length of a diagonal (the distance between two opposite vertices), what is the perimeter of the square? If you find the length of AC, you will have the first piece!


-Patrick

Hello Patrick,

Thanks a lot for the excellent explanation (and visuals).

Best Regards,
Sri

GMATGuruNY wrote:

gmattesttaker2 wrote: Points A and C have (x,y) coordinates (-1,3) and (2,-1) respectively, and are
opposite vertices of square ABCD, what is the perimeter of square ABCD?

When no figure is given, there is a likely reason:
If a figure were given, the problem would be too EASY.
Implication:
Be careful when drawing.
The problem probably contains a TWIST.


As the figure above shows, AC is the hypotenuse of a 3-4-5 triangle.
Thus, AC = 5.

AC is also the diagonal of square ABCD.
The diagonal of a square = s√2.
Since AC = 5, we get:
s√2 = 5.
s = 5/√2.

Thus:
p = 4s = 4 * (5/√2) = (4*5*√2) / (√2*√2) = (20√2)/2 = [spoiler]10√2[/spoiler].

Hello Mitch,

Thanks a lot for the excellent visual and explanation.

Best Regards,
Sri

The two points are = A(-1,3) and C(2,-1)

Using distance formula the length of diagonal AC is =
=> sqrt(square(-1-2)+square(3+1))= sqrt(square(-3)+square(4)) = sqrt(9+16) = sqrt(25) = 5

Now, if we draw the square ABCD, ABC forms a right angled triangle.

Therefore, as per Pythagoras theorem,
=> square AC = square AB + square BC
OR square 5 = square AB + square BC
OR 25 = square AB + square BC
As ABCD is a square therefore AB = BC,therefore,
=> 25 = square AB + square AB
OR 25 = 2 * square AB
OR 25/2 = square AB
OR sqrt(25/2)= AB
OR 5/(sqrt 2)= AB

Now as denominator has a number in square root, therefore, we will rationalize the number by multiplying the numerator and denominator by "sqrt (2)"

Hence we get,

=> (5 sqrt 2)/(sqrt 2 sqrt 2) = AB = (5 sqrt 2)/2

As one side is AB, and all sides of a square are always equal, therefore, perimeter is "4AB"

=> 4 AB = 4 (5 sqrt 2)/2 = "10 sqrt 2" units Ans.

---------------------------------
Er. Richa Yadav

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