Prove that the points A(1,7) B(4,2) …

Let A(1, 7), B(4, 2), C(-1, -1) and D(-4, 4) 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 both its diagonals should also be equal. Now,
{tex}A B = \sqrt { ( 1 - 4 ) ^ { 2 } + ( 7 - 2 ) ^ { 2 } } = \sqrt { 9 + 25 } = \sqrt { 34 }{/tex}
{tex}B C = \sqrt { ( 4 + 1 ) ^ { 2 } + ( 2 + 1 ) ^ { 2 } } = \sqrt { 25 + 9 } = \sqrt { 34 }{/tex}
{tex}C D = \sqrt { ( - 1 + 4 ) ^ { 2 } + ( - 1 - 4 ) ^ { 2 } } = \sqrt { 9 + 25 } = \sqrt { 34 }{/tex}
{tex}D A = \sqrt { ( 1 + 4 ) ^ { 2 } + ( 7 - 4 ) ^ { 2 } } = \sqrt { 25 + 9 } = \sqrt { 34 }{/tex}
{tex}A C = \sqrt { ( 1 + 1 ) ^ { 2 } + ( 7 + 1 ) ^ { 2 } } = \sqrt { 4 + 64 } = \sqrt { 68 }{/tex}
{tex}B D = \sqrt { ( 4 + 4 ) ^ { 2 } + ( 2 - 4 ) ^ { 2 } } = \sqrt { 64 + 4 } = \sqrt { 68 }{/tex}
Since, AB = BC = CD = DA and AC = BD, all the four sides of the quadrilateral ABCD are equal and its diagonals AC and BD are also equal. Therefore, ABCD is a square.