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

eramsby1O1O Jun 1, 2024

#1**+1 **

To start, if all sides of a quadrilateral are equal and the diaganols are equal, then we defintely have a square.

The square of diagonal of a square is equal to 2 times the square of the sidelegnth. We have

\(2PQ^2 = (\sqrt2)^2 \\ 2PQ^2 = 2 \\ PQ^2 = 1 \\ PQ = 1\)

The perimeter is just \(1+1+1+1 = 4\)

So 4 is the answer.

(We could check. The diagonal of a square with sidelength 1 is actually sqrt2, so we are clear!)

Thanks! :)

NotThatSmart Jun 1, 2024

#1**+1 **

Best Answer

To start, if all sides of a quadrilateral are equal and the diaganols are equal, then we defintely have a square.

The square of diagonal of a square is equal to 2 times the square of the sidelegnth. We have

\(2PQ^2 = (\sqrt2)^2 \\ 2PQ^2 = 2 \\ PQ^2 = 1 \\ PQ = 1\)

The perimeter is just \(1+1+1+1 = 4\)

So 4 is the answer.

(We could check. The diagonal of a square with sidelength 1 is actually sqrt2, so we are clear!)

Thanks! :)

NotThatSmart Jun 1, 2024