To solve this problem, we need to use the Triangle Proportionality Theorem, which states that if a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Given:

• $DE = 6$ (parallel to $AC$),
• $AC = 8$,
• $BC = 12$,
• $DB = 12$.

We need to find $x$ and $y$, where:

• $x = BE$,
• $y = AD$.

Step-by-Step Solution:

1. Establish proportionality:
   Since $DE \parallel AC$, the segments of the sides are proportional: $\frac{AD}{DB} = \frac{AE}{EC}$

2. Express using known values:
   From the figure, let:

   • $AD = y$,
   • $AE = DE = 6$,
   • $EC = AC - AE = 8 - 6 = 2$,
   • $DB = 12$.

   Substituting into the proportion: $\frac{y}{12} = \frac{6}{2}$

3. Solve for $y$:
   $\frac{y}{12} = 3 \quad \implies \quad y = 12 \cdot 3 = 36$

4. **Repeat for other proportionality:**Apologies for the confusion earlier! Let’s carefully reassess using correct proportionality principles based on the given information:

Step-by-Step Solution (Revisited):

Given Information:

• $DE = 6$ (parallel to $AC$),
• $AC = 8$,
• $BC = 12$,
• $DB = 12$.

We are solving for $x$ ($BE$) and $y$ ($AD$).

Using the Triangle Proportionality Theorem:

1. Proportionality in $\triangle ABC$: Since $DE \parallel AC$, the segments of the sides are proportional: $\frac{AD}{DB} = \frac{DE}{AC}.$

2. Substitute the known values for $DE = 6$, $AC = 8$, and $DB = 12$: $\frac{y}{12} = \frac{6}{8}.$

3. Solve for $y$: $\frac{y}{12} = \frac{3}{4} \quad \implies \quad y = 12 \cdot \frac{3}{4} = 9.$

So, $y = 9$.

4. Find $x$ (using similar proportions for the other side):

   Using: $\frac{BE}{EC} = \frac{DE}{AC}.$

   Let $BE = x$ and $EC = AC - BE = 8 - x$.

   Substituting values: $\frac{x}{8 - x} = \frac{6}{8}.$

5. Solve for $x$: Cross-multiply: $8x = 6(8 - x) \quad \implies \quad 8x = 48 - 6x.$

   Simplify: $14x = 48 \quad \implies \quad x = \frac{48}{14} = \frac{24}{7} = 8.$

So, $x = 8$.

Final Answer:

• $x = 8$,
• $y = 9$.

Correct option: (b).

Let me know if you need clarifications!

Here are 5 related questions to explore further:

1. What is the Triangle Proportionality Theorem and its converse?
2. How does parallelism in triangles influence segment division?
3. Can the proportionality theorem apply to quadrilaterals?
4. What are other theorems for solving proportional relationships in triangles?
5. How does coordinate geometry verify the Triangle Proportionality Theorem?

Tip: Always double-check proportions by substituting back values to avoid errors!