 ## ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA

[question] Question. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that: (i) $S R \| A C$ and $S R=\frac{1}{2} A C$ (ii) $P Q=S R$ (iii) $\mathrm{PQRS}$ is a parallelogram. [/question] [solution] Solution: (i) In ΔADC, S and R are the mid-points of sides AD and CD respectively. In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is ha...

## ABCD is a trapezium in which AB || CD and AD = BC

[question] Question. ABCD is a trapezium in which AB || CD and AD = BC (see the given figure). Show that (i) $\angle \mathrm{A}=\angle \mathrm{B}$ (ii) $\angle \mathrm{C}=\angle \mathrm{D}$ (iii) $\triangle \mathrm{ABC} \cong \triangle \mathrm{BAD}$ (iv) diagonal $A C=$ diagonal $B D$ [/question] [solution] Solution: Let us extend AB. Then, draw a line through C, which is parallel to AD, intersecting AE at point E. It is clear that AECD is a parallelogram. (i) $A D=C E$ (Opposite sides of parall...