1. **Problem statement:** Verify that the quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS with vertices P(0, 6), Q(-6, -2), R(2, -4), and S(4, 2) is a parallelogram. 2. **Step 1: Find the midpoints of each side of PQRS.** - Midpoint of PQ: $$M_1 = \left(\frac{0 + (-6)}{2}, \frac{6 + (-2)}{2}\right) = (-3, 2)$$ - Midpoint of QR: $$M_2 = \left(\frac{-6 + 2}{2}, \frac{-2 + (-4)}{2}\right) = (-2, -3)$$ - Midpoint of RS: $$M_3 = \left(\frac{2 + 4}{2}, \frac{-4 + 2}{2}\right) = (3, -1)$$ - Midpoint of SP: $$M_4 = \left(\frac{4 + 0}{2}, \frac{2 + 6}{2}\right) = (2, 4)$$ 3. **Step 2: Show that quadrilateral formed by points M1, M2, M3, M4 is a parallelogram by proving opposite sides are parallel.** - Vector $\overrightarrow{M_1M_2} = ( -2 - (-3), -3 - 2 ) = (1, -5)$ - Vector $\overrightarrow{M_3M_4} = ( 2 - 3, 4 - (-1) ) = (-1, 5)$ Check if $\overrightarrow{M_1M_2}$ is parallel to $\overrightarrow{M_3M_4}$: $$\text{Are } (1, -5) \text{ and } (-1, 5) \text{ scalar multiples?}$$ Yes, $(1, -5) = -1 \times (-1, 5)$, so they are parallel. - Vector $\overrightarrow{M_2M_3} = (3 - (-2), -1 - (-3)) = (5, 2)$ - Vector $\overrightarrow{M_4M_1} = (-3 - 2, 2 - 4) = (-5, -2)$ Check if $\overrightarrow{M_2M_3}$ is parallel to $\overrightarrow{M_4M_1}$: $$\text{Are } (5, 2) \text{ and } (-5, -2) \text{ scalar multiples?}$$ Yes, $(5, 2) = -1 \times (-5, -2)$, so they are parallel. 4. **Step 3: Conclusion** Since both pairs of opposite sides are parallel, the quadrilateral formed by joining the midpoints of PQRS is a parallelogram. **Final answer:** The quadrilateral formed by joining the midpoints of the sides of PQRS is a parallelogram.