Question

ABCD is a quadrilateral p,q,r,s are the mid points of the sides AB,BC,CD,DA .AC is the diagonal show that SR//AC and SR=1/2AC. PQ=SR PQRS is a parallelogram​

Answer 1

(i) $\large\tt\red{In\:∆DAC,}$

$\small\tt{S\:and\:R\:are\:the\:mid-points\:of\:side\:AD\:and\:DC}$

$\large\tt\purple{Thus,By\:mid-point\:theorem,}$

$\small\tt{SR||AC\:and\:SR=\frac{1}{2}AC}$......(1)

━━━━━━━━━━━━━━━

(ii) $\large\tt\red{In\:∆ABC,}$

$\small\tt{P\:and\:Q\:are\:the\:mid-points\:of\:side\:AB\:and\:BC}$

$\large\tt\purple{Thus,By\:mid-point\:theorem,}$

$\small\tt{PQ||AC\:and\:PQ=\frac{1}{2}AC}$......(2)

━━━━━━━━━━━━━━━

$\therefore$$\large\tt\green{From\:(1)\:and\:(2)}$

$\huge\tt\orange{PQ||SR}$

━━━━━━━━━━━━━━━

(iii) $\large\tt\pink{SR||AC\:and\:PQ||AC}$

Thus, PQRS is a ||gm because opposite sides of quadrilateral PQRS is equal and parallel.