In the following figures, PQ = LM, Q is the midpoint of PR and M is the midpoint of LN. Using the axioms, prove that PR = LN.

If S is the mid-point of side QR of a DeltaPQR , then prove that PQ+PR=2PS .

In the following figure, if AB = CD, prove that AC = BD. State the axioms used in proving the result.

In the following figure, t is the transversal of l and m. If angleAPQ=72^(@) and angleDQF=108^(@) , prove that l||m .

In a quadrilateral P Q R S , vec P Q= vec a , vec Q R = vec b , vec S P= vec a- vec b ,M is the midpoint of vec Q Ra n dX is a point on S M such that S X=4/5S Mdot Prove that P ,Xa n dR are collinear.

In the given figure, AB = 36 cm and M is the midpoint of AB. Semicircles are drawn on AB, AM and MB as diameters. A circle with centre C touches all the three circles. Find the area of shaded region. [Hint : Take O as the centre of semicircle on diameter AM. Take the radius of the circle with centre C as r. Then , OC = (9 + r)cm, OM = 9 cm and CM = (18-r) cm. Now, use Pythagoras theorem and find r = 6 cm.]

In the adjacent figure ABCD is a parallelogram P and Q are the midpoints of sides AB and DC respectively. Show that PBCQ is also a parallelogram.

In the figure shown, the spring are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. Rotational SHM of the rod (Mass m , length L ) will occur only if-

Let O A C B be a parallelogram with O at the origin and O C a diagonal. Let D be the midpoint of O Adot using vector methods prove that B Da n dC O intersect in the same ratio. Determine this ratio.

ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that CQ" ||" AP and also AC bisects PQ (see figure).

Find the area of the shaded region in the given figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.