`overlineA xx overline B = overlineB xx overline A` or not?

If overline (x_(1)) " and " overline (x_(2)) are the means of two distributions such that overline (x_(1)) lt overline (x_(2)) " and " overline (x) is the mean of the combined distriubtion, then

P is a point on the line - segment overline(AB) such that overline(AP)=overline(3PB) , if the coordinates of A and B are (3,-4) and (-5,2) respectively , find the coordinates of P.

A(-1,5),B(3,1)and C(5,7) are the vertices of the DeltaABC . If D ,E and F are the mid - points of the sides overline(BC),overline(CA)andoverline(AB) respectively , find the area of DeltaDEF . Show also that , DeltaABC=4DeltaDEF .

If the coordinates of the vertices of a triangle ABC be (3,0), (0,6)and (6,9)and if D and E respectively divide overline(AB) and overline(AC) internally in the ratio 1:2 , then show that the area of DeltaABC=9 x the area of DeltaADE .

D,E,F are the mid - points of the sides overline(BC),overline(CA)andoverline(AB) respectively of the triangle ABC , using coordinate geometry show that , 3(BC^(2)+CA^(2)+AB^(2))=4(AD^(2)+BE^(2)+CF^(2)) .

The coordinates of the points A,B,C and D are (-2,3), (8,9), (0,4) and (3,0) respectively . Find the ratio in which the line -segment overline(AB) is divided by the line -segment overline(CD) .

Let PQR be a triangle . Let veca=overline(QR),vecb = overline(RP) and vecc= overline(PQ).if |veca|=12, |vecb|=4sqrt3and vecb.vecc= 24 then which of the following is (are) true ?

At 27^(@)C , average translational kinetic energies of the molecules in 8g of CH_(4),8g of O_(2) and 8g of He are overline(in_(1)),overline(in_(2)) and overline(in_(3)) respectively and total kinetic energies of the molecules in these gases are E_(1),E_(2) and E_(3) respectively. which of the following is true-