1. **Problem Statement:** ABCD is a rhombus and P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Show that quadrilateral PQRS is a rectangle. 2. **Formula and Important Rules:** - In a rhombus, all sides are equal and opposite sides are parallel. - Midpoint formula: If $M$ is midpoint of segment with endpoints $X(x_1,y_1)$ and $Y(x_2,y_2)$, then $M=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. - To prove PQRS is a rectangle, we need to show that all angles are right angles, which can be done by showing adjacent sides are perpendicular. 3. **Intermediate Work:** - Let coordinates of rhombus ABCD be $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, $D(x_4,y_4)$. - Midpoints: $$P=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right),\quad Q=\left(\frac{x_2+x_3}{2}, \frac{y_2+y_3}{2}\right),$$ $$R=\left(\frac{x_3+x_4}{2}, \frac{y_3+y_4}{2}\right),\quad S=\left(\frac{x_4+x_1}{2}, \frac{y_4+y_1}{2}\right)$$ - Vector $\overrightarrow{PQ} = Q - P = \left(\frac{x_3 - x_1}{2}, \frac{y_3 - y_1}{2}\right)$ - Vector $\overrightarrow{QR} = R - Q = \left(\frac{x_4 - x_2}{2}, \frac{y_4 - y_2}{2}\right)$ 4. **Show Perpendicularity:** - Dot product $\overrightarrow{PQ} \cdot \overrightarrow{QR} = 0$ means vectors are perpendicular. - Calculate: $$\overrightarrow{PQ} \cdot \overrightarrow{QR} = \frac{(x_3 - x_1)(x_4 - x_2) + (y_3 - y_1)(y_4 - y_2)}{4}$$ - Since ABCD is a rhombus, $\overrightarrow{AC}$ is perpendicular to $\overrightarrow{BD}$, so the above dot product is zero. 5. **Conclusion:** - Since adjacent sides of PQRS are perpendicular, PQRS is a rectangle. **Final answer:** Quadrilateral PQRS formed by joining midpoints of sides of rhombus ABCD is a rectangle.