Two circles of radii 12 cm and 5 cm lie wholly outside each other, and the length of their direct common tangent is 24 cm. How far apart are their centres?

The direct common tangent satisfies $t$ = √$(d^2-(r_1-r_2{)}^2)$, so run it backwards: $d^2=t^2+(r_1-r_2{)}^2=24^2+7^2=576+49=625$, giving $d=25$ cm. The answer is consistent, since 25 is comfortably more than 12 + 5 = 17 and the circles really do lie apart. The trap √865 uses the SUM of the radii, $24^2+17^2$, which belongs to the transverse tangent instead; 17 cm is the sum of the radii itself, the condition for touching externally rather than a distance to be found; and 31 cm adds 24 and 7 without squaring.