A bit of vector algebra. But really only the pythagorean theorem.

We can find the vectors for each diagonal by subtracting startpoint from endpoint, so

AC = (12,-2) - (0,7) = (12-0, -2-7) = (12,-9)

BD = (7,8) - (1,0) = (7-1, 8-0) = (6,8)

We can find the length of each of these by taking the square of each vector component, add

them togehter and then taking the square root of this sum.

So the length of AC, denoted |AC| = \(\sqrt{12\times12 + (-9)\times(-9)} = \sqrt{144 + 81} = \sqrt{225} = 15\)

Similarly, the length of BD, |BD| = \(\sqrt{6\times6 + (8)\times(8)} = \sqrt{36 + 64} = \sqrt{100} = 10\)

The rule for finding the Area of the kite is 1/2 x |AC| x |BD| = 1/2 x 15 x 10 = 1/2 x 150 = 75

Hope this helps.