If the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square then what is the length of the diagonal of square?

The points (1, 7), (4, 2), (-1, -1) and (-4, 4) form a square. In a square, all four sides are equal in length, and both diagonals are equal in length. The diagonals are longer than the sides.

Label the points as:

A = (1, 7)

B = (4, 2)

C = (-1, -1)

D = (-4, 4)

Using the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$:

Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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Distance from A(1, 7) to B(4, 2):

$= \sqrt{(4-1)^2 + (2-7)^2}$

$= \sqrt{3^2 + (-5)^2}$

$= \sqrt{9 + 25}$

$= \sqrt{34}$

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Distance from A(1, 7) to C(-1, -1):

$= \sqrt{(-1-1)^2 + (-1-7)^2}$

$= \sqrt{(-2)^2 + (-8)^2}$

$= \sqrt{4 + 64}$

$= \sqrt{68}$

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Distance from B(4, 2) to C(-1, -1):

$= \sqrt{(-1-4)^2 + (-1-2)^2}$

$= \sqrt{(-5)^2 + (-3)^2}$

$= \sqrt{25 + 9}$

$= \sqrt{34}$

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Calculating all distances between the four points yields:

• Four distances equal to $\sqrt{34}$ (the sides of the square)
• Two distances equal to $\sqrt{68}$ (the diagonals of the square)

Therefore, the length of the diagonal of the square is $\sqrt{68}$ units.