The coordinates of the points P, Q, R and S are shown in the figure ab

Question

https://gmatclub.com/forum/download/file.php?id=39271 The coordinates of the points P, Q, R and S are shown in the figure above. If PQ and RS are the diagonals of a square, what is the area of the square? (A) 2√2 (B) 4 (C) 4√2 (D) 8 (E) 8√2 2017-12-19_0837_002.png

pushpitkc · Answer

If the figure is a square, diagonals are equal, length of which is 4.

If the length of the diagonal is 4, the length of the side is 4/√2 = 2√2

Area of the square is (2√2)^2 or 8(Option D)

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exc4libur · Answer

Distance between two points:  \sqrt{(x2-x1)^2+(y2-y1)^2} = \sqrt{(5-1)^2+(5-5)^2}=√16=4

Side of square using the diagonal:  diagonal = side*√2; (4) = side*√2; side = \frac{4}{√2}

Area of square:  side^2=(\frac{4}{√2})^2=\frac{6}{2}=8

(D) is the answer.

nahid78 · Answer

If the diagonals of a square is a and b, the the area is (a*b)/2

destinyawaits · Answer

I found the area of each triangle... (1/2 * base * height) * 4 (number of triangles in square)

testcracker · Answer

lets find the distance between SP, or PR, or RQ, or QS

the distance between any of them is √8

so area of the square is 8 = answer choice D

thanks

generis · Answer

Three ways to solve, below, are all based on the length of the diagonal.

Diagonal = length of line segment RS (or PQ, in which case use x-coordinates instead):
 (y_2 - y_1) = (7 - 3) = 4 = d

1. Area from diagonal alone:
  A= \frac{d^2}{2}  
Area of square =   \frac{d^2}{2}=\frac{4^2}{2}=\frac{16}{2}=8

2. Area from  s^2 , calculated from diagonal length
 s\sqrt{2} = d 
 s=\frac{d}{\sqrt{2}} 
 s=\frac{4}{\sqrt{2}} 
 Area = s^2 
 Area = (\frac{4}{\sqrt{2}}*\frac{4}{\sqrt{2}})= \frac{16}{2} = 8

3. Area from sides and Pythagorean theorem

A square's diagonal is the hypotenuse of an isosceles right triangle. Use Pythagorean theorem:
 s^2 + s^2 = d^2 
 2s^2 = 4^2 
 s^2 = \frac{4^2}{2} 
 Area = s^2 = \frac{4^2}{2}=\frac{16}{2}= 8

ANSWER D

*Note: obvious, but often overlooked. Do not solve for  s  in #3. Once you have  s^2 , you have area.

ashisplb · Answer

D1- diagonal one
D2=diagonal2
D1=√(5-1)^2 +(5-5)^2=4
D2=√(3-3)^2+(7-3)^2=4

Area=(D1*D2)/2=(4*4)/2=8

Gladiator59 · Answer

Since it is  GIVEN  to be a square - you don't even need to calculate the length of both the diagonals... since the diagonals of a square are equal in length.

Could save a few precious seconds...

AkshdeepS · Answer

Distance between PR =  \sqrt{(3-1)^2 + (7-5)^2}

PR =  \sqrt{4 + 4}

PR =  2*\sqrt{2}

Distance between RQ =  \sqrt{(3-5)^2 + (7-5)^2}

RQ =  \sqrt{(-2)^2 + (2)^2}

RQ =  2*\sqrt{2}

Area of square = PR * RQ

2\sqrt{2} * 2\sqrt{2}

Area = 8

Hence (D)

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The coordinates of the points P, Q, R and S are shown in the figure ab

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The coordinates of the points P, Q, R and S are shown in the figure ab

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