Prove that the points (4,5),(7,6),(6,3)and(3,2)are the vertices of a parallelogram.is it a rectangle

Prove that the points (4,5),(7,6),(6,3)and(3,2)are the vertices of a p

1.	Prove that the points (4,5),(7,6),(6,3)and(3,2)are the vertices of a parallelogram.is it a rectangle
Answer» Let A(4, 5), B(7, 6), C(6, 3) and D(3, 2) be the given points.And, P the point of intersection of AC and BD.Coordinates of the mid-point of Ac are\xa0{tex}\\left( \\frac { 4 + 6 } { 2 } , \\frac { 5 + 3 } { 2 } \\right) = ( 5,4 ){/tex}Coordinates of the mid-point of BD are\xa0{tex}\\left( \\frac { 7 + 3 } { 2 } , \\frac { 6 + 2 } { 2 } \\right) = ( 5,4 ){/tex}Thus, AC and BD have the same mid-point.Hence, ABCD is a parallelogram.Now we shall see whether ABCD is a rectangle.We have,{tex}A C = \\sqrt { ( 6 - 4 ) ^ { 2 } + ( 3 - 5 ) ^ { 2 } }{/tex}{tex}\\Rightarrow \\quad A C = \\sqrt { 4 + 4 }{/tex}{tex}\\Rightarrow \\quad A C = \\sqrt { 8 }{/tex}And,\xa0{tex}B D = \\sqrt { ( 7 - 3 ) ^ { 2 } + ( 6 - 2 ) ^ { 2 } }{/tex}{tex}\\Rightarrow \\quad B D = \\sqrt { 16 + 16 }{/tex}{tex}\\Rightarrow \\quad B D = \\sqrt { 32 }{/tex}Since, AC\xa0{tex}\\neq{/tex}\xa0BDSo, ABCD is not a rectangle.