1. **State the problem:** We need to show that the quadrilateral PATR with vertices $P(-3,3)$, $A(3,2)$, $T(-4,-3)$, and $R(2,-4)$ is a parallelogram by finding the lengths of its sides and verifying properties of parallelograms. 2. **Formula for distance between two points:** The distance between points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate each side length:** - Side $PA$: $$PA = \sqrt{(3 - (-3))^2 + (2 - 3)^2} = \sqrt{(6)^2 + (-1)^2} = \sqrt{36 + 1} = \sqrt{37}$$ - Side $TP$: $$TP = \sqrt{(-3 - (-4))^2 + (3 - (-3))^2} = \sqrt{(1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37}$$ - Side $TR$: $$TR = \sqrt{(2 - (-4))^2 + (-4 - (-3))^2} = \sqrt{(6)^2 + (-1)^2} = \sqrt{36 + 1} = \sqrt{37}$$ - Side $RA$: $$RA = \sqrt{(3 - 2)^2 + (2 - (-4))^2} = \sqrt{(1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37}$$ 4. **Interpretation:** All four sides have length $\sqrt{37}$. Since opposite sides are equal in length, this is a key property of parallelograms. 5. **Conclusion:** The quadrilateral PATR has opposite sides equal, so it is a parallelogram. Final answer: $$PA = TP = TR = RA = \sqrt{37}$$