\(P\) is the point \((-3, 5)\) and \(Q\) is the point \((9, -2)\). Find the length of \(PQ\), correct to 1 decimal place.

The distance between two points comes from Pythagoras' theorem applied to the right-angled triangle whose horizontal and vertical sides are the differences in the coordinates:

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

From \(P(-3, 5)\) to \(Q(9, -2)\), the horizontal change is \(9 - (-3) = 12\) and the vertical change is \(-2 - 5 = -7\). Squaring removes the sign, so only the sizes matter:

\(PQ = \sqrt{12^2 + 7^2} = \sqrt{144 + 49} = \sqrt{193} = 13.892\ldots = 13.9\) (1 d.p.) [B1]

The step where marks are most often lost is the horizontal difference: subtracting a negative gives \(9 + 3 = 12\), not 6. Because both differences are squared, it makes no difference whether you work from \(P\) to \(Q\) or from \(Q\) to \(P\). A rough check confirms the size: the distance must be longer than either separate side, so it must exceed 12, and it must be less than \(12 + 7 = 19\), the path taken by going across then up. A value of 13.9 sits comfortably between them.