Prove that the quadrilatral formed by joining the mid points of the sides of a quadrilateral is a parallelogram.

Prove that the Iine joining the midpoints of the two sides of a triangle is parallel to the third side and half of its length.

Calculate the length of sides of quadrilateral. .

The coordinates of the vertices of a quadrilateral taken in order are '(2,1)', '(5,3),(8,7),(4,9)'. i) Find the coordinates of the midpoints of all four sides. ii) Prove that the quadrilateral got by joining these mid points is a parallelogram.

Prove that if both pairs of opposite sides of a quadrilateral are equal, then it is a parallelogram.

The coordinates of the verticesofa quadrilateral, taken in order, are (2,1), (5,3), (8,7), (4,9).Prove that the quadrilateral got by joining these midpoints is a parallelogram.

Consider the following quadrilateral ABCD in which P,Q,R,S are the mid points of the sides. Show that PQRS is a parallelogram.

Prove that the quadrilateral obtained by joining any two alternate vertices of a regular pentagon is cyclic.

A square is drawn by joining the mid points of the sides of a square. A third square is drawn inside the second square in the same way and the process is continued infinitely. If the side of the square is 10 cms, find the sum of the areas of all the squares so formed.

Let S_(1) be a square of side is 5 cm. Another square S_(2) is drawn by joining the mid-points of the sides of S_(1) . Square S_(3) is drawn by joining the mid-points of the sides S_(2) and so on. Then, (area of S_(1) + area of S_(2)+…+ area of S_(10) ) is a) 25(1-1/(2^(10)))cm^(2) b) 50(1-1/(2^(10)))cm^(2) c) 2(1-1/(2^(10)))cm^(2) d) 1-1/(2^(10))cm^(2)