If the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices o…

The correct answer is Option (2) → \(\sqrt{68}\) units

The vertices are given in order round the square, so the 1st and 3rd points, and the 2nd and 4th, are opposite vertices; a diagonal joins them. Its length comes from the distance formula, \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).

The sides (equal, as a square needs):

• Side from (1, 7) to (4, 2): \(\sqrt{3^2+5^2}=\sqrt{34}\)

• Side from (4, 2) to (−1, −1): \(\sqrt{5^2+3^2}=\sqrt{34}\), the same

The diagonals (joining opposite vertices):

• Diagonal from (1, 7) to (−1, −1): \(\sqrt{2^2+8^2}=\sqrt{4+64}=\sqrt{68}\)

• Other diagonal from (4, 2) to (−4, 4): \(\sqrt{8^2+2^2}=\sqrt{68}\), equal to the first

The value \(\sqrt{34}\) in Option (4) is the side of the square; the diagonal is \(\sqrt{2}\) times the side, \(\sqrt{34}\times\sqrt{2}=\sqrt{68}\).