and its diagonals AC and BD are also equal. Thereore, ABCD is a square. Alternative Solution : We find the four sides and one diagonal, say, AC as above. Here AD + DC = + = = AC .
Therefore, by the converse of Pythagoras theorem, D = °. A quadrilateral with all four sides equal and one angle ° is a square. So, ABCD is a square. Example : Fig.
. shows the arrangement of desks in a classroom. Ashima, Bharti and Camella are seated at A( , ), B( , ) and C( , ) respectively. Do you think they are seated in a line?
Give reasons for your answer. Fig. . Solution : Using the distance formula, we have AB = ( ) ( ) BC = ( – ) ( – ) AC = ( – ) ( – ) Since, AB + BC = AC, we can say that the points A, B and C are collinear.
Therefore, they are seated in a line. Example : Find a relation between x and y such that the point ( x , y ) is equidistant from the points ( , ) and ( , ). Solution : Let P( x , y ) be equidistant from the points A( , ) and B( , ). We are given that AP = BP.
So, AP = BP ( x – ) + ( y – ) = ( x – ) + ( y – ) x – x + + y – y + = x – x + + y – y + x – y = which is the required relation. Remark : Note that the graph of the equation x – y = is a line. From your earlier studies, you know that a point which is equidistant from A and B lies on the perpendicular