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# Geometry

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

Jun 1, 2024

#1
+1804
+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! :)

Jun 1, 2024

#1
+1804
+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