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

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Posts: 52161
The coordinates of the points P, Q, R and S are shown in the figure ab  [#permalink]

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18 Dec 2017, 20:46
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Difficulty:

5% (low)

Question Stats:

81% (01:18) correct 19% (01:30) wrong based on 143 sessions

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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 [ 10.19 KiB | Viewed 1515 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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18 Dec 2017, 20:54
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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19 Dec 2017, 06:32
Bunuel wrote:

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: $$side^2=(\frac{4}{√2})^2=\frac{6}{2}=8$$

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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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15 Jan 2018, 09:27
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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15 Jan 2018, 19:48
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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23 Feb 2018, 08:57
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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23 Feb 2018, 22:41
Bunuel wrote:

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:
$$s^2 + s^2 = d^2$$
$$2s^2 = 4^2$$
$$s^2 = \frac{4^2}{2}$$
$$Area = s^2 = \frac{4^2}{2}=\frac{16}{2}= 8$$

*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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12 Mar 2018, 01:35
Bunuel wrote:

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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12 Mar 2018, 02:04
ashisplb wrote:
Bunuel wrote:

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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11 Apr 2018, 08:41
Bunuel wrote:

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, 08:41
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