x , y ) and Q( x , y ) is PQ = – + – which is called the distance formula . Remarks : . In particular, the distance of a point P( x , y ) from the origin O( , ) is given by OP = . .
We can also write, PQ = . (Why?) Example : Do the points ( , ), (– , – ) and ( , ) form a triangle? If so, name the type of triangle formed. Solution : Let us apply the distance formula to find the distances PQ, QR and PR, where P( , ), Q(– , – ) and R( , ) are the given points.
We have PQ = ( ) ( ) = . (approx.) QR = (– – ) (– – ) (– ) (– ) = . (approx.) PR = ( – ) ( – ) ( ) = . (approx.) Since the sum of any two of these distances is greater than the third distance, therefore, the points P, Q and R form a triangle.
Fig. . Also, PQ + PR = QR , by the converse of Pythagoras theorem, we have P = °. Therefore, PQR is a right triangle.
Example : Show that the points ( , ), ( , ), (– , – ) and (– , ) are the vertices of a square. Solution : Let A( , ), B( , ), C(– , – ) and D(– , ) be the given points. One way of showing that ABCD is a square is to use the property that all its sides should be equal and both its digonals should also be equal. Now, AB = ( – ) ( ) BC = ( ) ( ) CD = (– ) (– – ) DA = ( ) ( – ) AC = ( ) ( ) BD = ( ) ( ) Since, AB = BC = CD = DA and AC = BD, all the four sides of the quadrilateral ABCD are equal