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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  [#permalink]

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New post 18 Dec 2017, 21:46
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A
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D
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Question Stats:

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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

Attachment:
2017-12-19_0837_002.png
2017-12-19_0837_002.png [ 10.19 KiB | Viewed 1413 times ]

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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 18 Dec 2017, 21:54
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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The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 19 Dec 2017, 07:32
Bunuel wrote:
Image
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

Attachment:
2017-12-19_0837_002.png


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.
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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 15 Jan 2018, 10:27
If the diagonals of a square is a and b, the the area is (a*b)/2
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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 15 Jan 2018, 20:48
I found the area of each triangle... (1/2 * base * height) * 4 (number of triangles in square)
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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 23 Feb 2018, 09:57
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
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The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 23 Feb 2018, 23:41
Bunuel wrote:
Image
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

Attachment:
2017-12-19_0837_002.png

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.
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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 12 Mar 2018, 02:35
Bunuel wrote:
Image
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

Attachment:
2017-12-19_0837_002.png



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
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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 12 Mar 2018, 03:04
ashisplb wrote:
Bunuel wrote:
Image
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

Attachment:
2017-12-19_0837_002.png



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


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...
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Re: The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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New post 11 Apr 2018, 09:41
Bunuel wrote:
Image
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

Attachment:
2017-12-19_0837_002.png


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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Re: The coordinates of the points P, Q, R and S are shown in the figure ab &nbs [#permalink] 11 Apr 2018, 09:41
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