To find the length of the leg $PQ,$ I can use the distance formula: $PQ = \sqrt{(6-0)^2 + (9-5)^2} = \sqrt{36 + 16} = \sqrt{52}.$ Similarly, to find the length of the leg $QR,$ I can use the distance formula again: $QR = \sqrt{(12-6)^2 + (0-9)^2} = \sqrt{36 + 81} = \sqrt{117}.$ Now I have the lengths of both legs, so I can plug them into the formula for the area: $A = \frac{1}{2} \cdot PQ \cdot QR = \frac{1}{2} \cdot \sqrt{52} \cdot \sqrt{117}.$ To simplify this expression, I can use the property that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ for any positive numbers $a$ and $b.$ So, $A = \frac{1}{2} \cdot \sqrt{52 \cdot 117} = \frac{1}{2} \cdot \sqrt{6084}.$ I notice that 6084 is divisible by 4, which is a perfect square, so I can write $A = \frac{1}{2} \cdot \sqrt{4 \cdot 1521} = \frac{1}{2} \cdot \sqrt{4} \cdot \sqrt{1521} = \frac{1}{2} \cdot 2 \cdot \sqrt{1521} = \sqrt{1521}.$ Finally, I can find the square root of 1521 by testing some integer factors. I see that 1521 is divisible by 3, so I can write $A = \sqrt{3 \cdot 507}.$ Then I see that 507 is also divisible by 3, so I can write $A = \sqrt{3 \cdot 3 \cdot 169}.$ Then I recognize that 169 is a perfect square, namely 13 squared, so I can write $A = \sqrt{3 \cdot 3 \cdot 13 \cdot 13} = \sqrt{9 \cdot 169} = \sqrt{9} \cdot \sqrt{169} = 3 \cdot 13 = 39.$ 
Let $P$ be the point $(0,5),$ let $Q$ be the point $(6,9),$ and let $R$ be the point $(12,0).$ Determine the area of right-angled $\triangle PQR.$