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# What is the area of Quadrilateral PQRS ? (1) PQ = QR = RS = SP = 1

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Senior DS Moderator
Joined: 27 Oct 2017
Posts: 1195
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
What is the area of Quadrilateral PQRS ? (1) PQ = QR = RS = SP = 1  [#permalink]

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11 May 2018, 20:16
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55% (hard)

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54% (01:26) correct 46% (01:03) wrong based on 57 sessions

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What is the area of Quadrilateral PQRS ?

(1) PQ = QR = RS = SP = 1
(2) PR =$$\sqrt{3}$$

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Joined: 16 Aug 2015
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Re: What is the area of Quadrilateral PQRS ? (1) PQ = QR = RS = SP = 1  [#permalink]

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12 May 2018, 05:47
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(1) is not sufficient alone. all sides equal means it can a square or a rhombus
(2) is not sufficient alone. the length of the diagonal alone cannot be used to determine the area.

Together they can be used to find the area.

Senior DS Moderator
Joined: 27 Oct 2017
Posts: 1195
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GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: What is the area of Quadrilateral PQRS ? (1) PQ = QR = RS = SP = 1  [#permalink]

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12 May 2018, 19:24
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(1) PQ = QR = RS = SP = 1
It shows the Quadrilateral PQRS is a Rhombus with sides 1, but since the area of rhombus can not be found only by knowing the length of the sides as it is not constant. In fact, its area will be maximum when included angle between sides is 90 deg( in that case, it will be a Square)
NOT SUFFICIENT.

(2) PR = root 3
Since Only a diagonal is known, no other information. It is NOT SUFFICIENT.

Now combining Statement 1 & 2,
we get PQRS is a Rhombus with side = 1, and diagonal = root 3.

Now visualize that the diagonal of a parallelogram divides the parallelogram into 2 triangles of equal area. So We need to find only the area of a triangle formed by 2 sides and a diagonal.
i) Since we know the sides = 1, and diagonal = root3
ii) all three sides are known, so unique triangle can be formed and hence the area of triangle.
iii) Now area of Rhombus will be twice the area of triangle.

SUFFICIENT.

(PS: Another Approach: As the diagonals of a Rhombus bisects each other perpendicularly, we cab find the other diagonal by Pythagoras theorem. And the Area would be (D1*D2)/2 = (Product of diagonals)/2)
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Re: What is the area of Quadrilateral PQRS ? (1) PQ = QR = RS = SP = 1 &nbs [#permalink] 12 May 2018, 19:24
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