let us use the understanding that you may have developed through this example to obtain the general formula. Consider any two points A( x , y ) and B( x , y ) and assume that P ( x , y ) divides AB internally in the ratio m : m , i.e., PA PB (see Fig. . ).
Draw AR, PS and BT perpendicular to the x -axis. Draw AQ and PC parallel to the x -axis. Then, by the AA similarity criterion, PAQ ~ BPC Therefore, PA AQ BP PC = PQ BC ( ) Now, AQ = RS = OS – OR = x – x PC = ST = OT – OS = x – x PQ = PS – QS = PS – AR = y – y BC = BT– CT = BT – PS = y – y Substituting these values in ( ), we get m = Taking m = , we get x = Similarly, taking m = , we get y = So, the coordinates of the point P( x , y ) which divides the line segment joining the points A( x , y ) and B( x , y ), internally, in the ratio m : m are ( ) This is known as the section formula . This can also be derived by drawing perpendiculars from A, P and B on the y -axis and proceeding as above.
If the ratio in which P divides AB is k : , then the coordinates of the point P will be kx ky Special Case : The mid-point of a line segment divides the line segment in the ratio : . Therefore, the coordinates of the mid-point P of the join of the points A( x , y ) and B( x , y ) is . Let us solve a few examples based on the section