D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.


We are given a triangle \( \Delta ABC \), and points \( D \) and \( E \) on sides \( AB \) and \( AC \), respectively, such that \( DE \parallel BC \) and \( \frac{AD}{DB} = \frac{7}{9} \). The lines \( BE \) and \( CD \) intersect at point \( F \). We are required to find the ratio of the areas of triangles \( \Delta DEF \) and \( \Delta CBF \).

### Step-by-Step Approach:

1. **Use of Basic Proportionality Theorem (Thales' Theorem):**

   Since \( DE \parallel BC \), by the Basic Proportionality Theorem, we know that:

   \[
   \frac{AD}{DB} = \frac{AE}{EC} = \frac{7}{9}
   \]

   Hence, \( \frac{AE}{EC} = \frac{7}{9} \).

2. **Ratio of areas of triangles with parallel sides:**

   Because \( DE \parallel BC \), the triangles \( \Delta ADE \) and \( \Delta ABC \) are similar, and the ratio of their areas will be the square of the ratio of their corresponding sides. Hence, the ratio of areas of triangles \( \Delta ADE \) and \( \Delta ABC \) is given by:

   \[
   \left( \frac{AD}{AB} \right)^2 = \left( \frac{7}{7+9} \right)^2 = \left( \frac{7}{16} \right)^2 = \frac{49}{256}
   \]

   Therefore, the area of \( \Delta ADE \) is \( \frac{49}{256} \) of the area of \( \Delta ABC \).

3. **Intersection properties of \( F \):**

   Since the lines \( BE \) and \( CD \) intersect at \( F \), \( F \) is known as the **centroid** of triangle \( \Delta BCD \). The centroid divides the area of a triangle into smaller triangles with specific area ratios, which we will use to find the required ratio.

4. **Using properties of the centroid:**

   The centroid divides each triangle into smaller triangles in the ratio 1:2. Thus, the areas of triangles \( \Delta DEF \) and \( \Delta CBF \) are in the same ratio as the corresponding sides of the triangles they lie in. Since \( DE \parallel BC \), the areas are proportional to the squares of the sides.

5. **Conclusion:**

   Based on the similarity of the triangles and the proportionality of areas, the ratio of the areas of triangles \( \Delta DEF \) and \( \Delta CBF \) is:

   \[
   \boxed{49:207}
   \]

This is the required ratio of the areas of triangles \( \Delta DEF \) and \( \Delta CBF \).

Would you like further clarification on any of the steps or need additional details?

### Related Questions:
1. How does the Basic Proportionality Theorem work in triangles?
2. What is the significance of the centroid in a triangle?
3. How do parallel lines affect area ratios in triangles?
4. Can we generalize this method to other similar problems?
5. How does the area ratio change if the ratio \( AD:DB \) is different?

### Tip:
Remember, when two triangles are similar, their areas are proportional to the squares of the corresponding sides. This concept is especially useful in geometry problems involving parallel lines and ratios.

D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F, then find the ratio of areas of ΔDEF and ΔCBF.

Solve a problem involving ΔABC where DE is parallel to BC and AD:DB = 7:9. Learn to apply the Basic Proportionality Theorem to find the ratio of areas of triangles ΔDEF and ΔCBF, using side length ratios and centroid properties. Suitable for students in Grades 10-12.

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