This problem involves proportional relationships created by parallel lines in a triangle-like configuration. Here’s how we solve step by step:

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Step 1: Understand the Geometry

• $\overline{PQ} \parallel \overline{MN}$: Parallel lines in a triangle create similar triangles by the Basic Proportionality Theorem (also known as Thales' Theorem).
• $\overline{OQ}$ and $\overline{ON}$ are the parts of the segment divided by the parallel lines.
• $\overline{PQ} = \overline{OQ} + 66$: Given.
• $\overline{ON} = 12$: Given.
• $\overline{MN} = 21$: Given.

From the Basic Proportionality Theorem, the ratios of the corresponding segments are equal: $\frac{\overline{OQ}}{\overline{ON}} = \frac{\overline{PQ}}{\overline{MN}}$

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Step 2: Substitution

From the problem, substitute the values into the proportion: $\frac{\overline{OQ}}{12} = \frac{\overline{OQ} + 66}{21}$

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Step 3: Solve the Proportion

Cross-multiply to eliminate the fractions: $21 \cdot \overline{OQ} = 12 \cdot (\overline{OQ} + 66)$

Distribute on the right-hand side: $21 \cdot \overline{OQ} = 12 \cdot \overline{OQ} + 12 \cdot 66$

Simplify: $21 \cdot \overline{OQ} = 12 \cdot \overline{OQ} + 792$

Combine like terms: $21 \cdot \overline{OQ} - 12 \cdot \overline{OQ} = 792$

$9 \cdot \overline{OQ} = 792$

Solve for $\overline{OQ}$: $\overline{OQ} = \frac{792}{9} = 88$

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Step 4: Verify

Now, substitute $\overline{OQ} = 88$ back into the given relationships:

• $\overline{PQ} = \overline{OQ} + 66 = 88 + 66 = 154$
• Check the proportion: $\frac{\overline{OQ}}{\overline{ON}} = \frac{\overline{PQ}}{\overline{MN}}$ $\frac{88}{12} = \frac{154}{21}$ Simplify both ratios: $\frac{88}{12} = \frac{22}{3}, \quad \frac{154}{21} = \frac{22}{3}$ The proportions are equal, so the solution is verified.

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Final Answer:

The length of $\overline{OQ}$ is: $\boxed{88}$