We are given that $P(0, 33)$ and $O(0, 0)$ are two points on a rectangular coordinate system, and the incenter of triangle $\triangle OPQ$ is $(12, 21)$. We are tasked with finding the distance between $O$ and $Q$.

Step 1: Understanding the Incenter Formula

The incenter of a triangle is the point where the angle bisectors meet, and it is located at the weighted average of the triangle's vertices based on the lengths of the opposite sides. The coordinates of the incenter are given by the following formula: $I_x = \frac{a x_A + b x_B + c x_C}{a + b + c}, \quad I_y = \frac{a y_A + b y_B + c y_C}{a + b + c}$ where $A(x_A, y_A), B(x_B, y_B), C(x_C, y_C)$ are the vertices of the triangle, and $a, b, c$ are the lengths of the sides opposite to vertices $A, B, C$, respectively.

For triangle $\triangle OPQ$, the vertices are $O(0, 0)$, $P(0, 33)$, and $Q(x_Q, y_Q)$. The incenter is $(12, 21)$.

Step 2: Side Lengths of the Triangle

• The distance $OP$ is simply the distance between $O(0, 0)$ and $P(0, 33)$: $OP = \sqrt{(0 - 0)^2 + (33 - 0)^2} = 33$

• Let the distance between $O$ and $Q(x_Q, y_Q)$ be denoted as $OQ$, and the distance between $P$ and $Q(x_Q, y_Q)$ as $PQ$.

Step 3: Incenter Coordinates

Using the formula for the incenter, we have: $12 = \frac{OQ \cdot x_Q + PQ \cdot 0 + 33 \cdot 0}{OQ + PQ + 33} = \frac{OQ \cdot x_Q}{OQ + PQ + 33}$ $21 = \frac{OQ \cdot y_Q + PQ \cdot y_Q + 33 \cdot 33}{OQ + PQ + 33}$ We now solve these equations.

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