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In quadrilateral $PQRS$, $PQ = QR = RS = SP$, $PR = \sqrt{2}$, and $QS = \sqrt{2}$.  Find the perimeter of $PQRS$.

 Dec 13, 2023
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We can solve this problem by analyzing the properties of the quadrilateral formed by the given information:

 

Sides: All four sides of the quadrilateral are equal, with length PQ=QR=RS=SP=2​. This suggests that the quadrilateral might be a regular polygon.

 

Diagonals: Both diagonals PR and QS have length 2​. In a square, the diagonals have length 2​ times the side length.

 

Considering these observations, we can conclude that the quadrilateral PQRS is a rhombus. A rhombus has all four sides equal and both diagonals perpendicular to each other.

 

Therefore, to find the perimeter of the rhombus, we just need to add the lengths of all four sides:

 

Perimeter of PQRS = PQ+QR+RS+SP=4⋅2​=42​​

 

So, the perimeter of the quadrilateral PQRS is 42​ units.

 Dec 13, 2023

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