In a right-angled triangle ABC with ∠B = 90°, the length of side AB = 8 units and the length of side BC = 6 units. Point D is on side AC such that AD: DC=2:3. Point E is on side BC such that BE: EC = 3: 4. If the area of triangle BDE is maximized, what is the ratio of the areas of triangles BDE and ABC?

Your first skill is: **Applying Geometric Concepts**
second skill Applying Propositional Logic

use these skills/topics to solve




We are tasked with finding the ratio of the areas of triangles \( \triangle BDE \) and \( \triangle ABC \), where certain conditions on the positions of points \( D \) and \( E \) apply. Let’s go through this step by step, applying geometric concepts and logical reasoning.

### 1. **Step 1: Analyze Triangle ABC**

- \( \triangle ABC \) is a right-angled triangle with \( \angle B = 90^\circ \).
- The lengths of sides are:
  - \( AB = 8 \) units
  - \( BC = 6 \) units
  - We can calculate the hypotenuse \( AC \) using the Pythagorean theorem:
  
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ units}.
\]

### 2. **Step 2: Set up the partition on sides AC and BC**

- Point \( D \) divides side \( AC \) in the ratio \( AD:DC = 2:3 \), meaning \( AD = \frac{2}{5}AC \) and \( DC = \frac{3}{5}AC \).
  - \( AD = \frac{2}{5} \times 10 = 4 \) units.
  - \( DC = \frac{3}{5} \times 10 = 6 \) units.

- Point \( E \) divides side \( BC \) in the ratio \( BE:EC = 3:4 \), meaning \( BE = \frac{3}{7}BC \) and \( EC = \frac{4}{7}BC \).
  - \( BE = \frac{3}{7} \times 6 = \frac{18}{7} \approx 2.57 \) units.
  - \( EC = \frac{4}{7} \times 6 = \frac{24}{7} \approx 3.43 \) units.

### 3. **Step 3: Area of triangle ABC**

The area of \( \triangle ABC \) is given by:

\[
\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 8 \times 6 = 24 \text{ square units}.
\]

### 4. **Step 4: Maximize the area of triangle BDE**

To maximize the area of triangle \( \triangle BDE \), we use the geometric fact that the area of any triangle is half the product of its base and height. Here, \( BE \) is part of side \( BC \), and point \( D \) divides side \( AC \). The area will depend on how these points divide the triangle, but it's maximized when both \( D \) and \( E \) lie in such a way that the triangle is stretched optimally within \( \triangle ABC \). 

### 5. **Step 5: Calculate the ratio of areas**

The area of \( \triangle BDE \) is a fraction of the area of \( \triangle ABC \), and this fraction depends on the positions of \( D \) and \( E \). Since \( D \) divides \( AC \) in the ratio 2:3 and \( E \) divides \( BC \) in the ratio 3:4, the area of \( \triangle BDE \) can be approximated by multiplying these ratios:

\[
\text{Ratio of areas} = \left( \frac{2}{5} \right) \times \left( \frac{3}{7} \right) = \frac{6}{35}.
\]

Thus, the ratio of the areas of triangles \( \triangle BDE \) and \( \triangle ABC \) is:

\[
\boxed{\frac{6}{35}}.
\]

### Final Result:
The ratio of the areas of triangles \( \triangle BDE \) and \( \triangle ABC \) is \( \frac{6}{35} \).

---

Would you like to explore any steps in more detail or clarify any concepts?

Here are 5 related questions you might want to consider:
1. How do we find the centroid of a triangle, and how could it help in area maximization?
2. What is the significance of the ratio \( AD:DC = 2:3 \) in determining the area?
3. How would the problem change if point \( E \) divided side \( BC \) in a different ratio?
4. Can the area of triangle BDE be further maximized by changing other parameters?
5. How can we apply similar triangle properties to calculate areas more directly?

**Tip:** When dealing with ratios in geometry, consider using area proportions based on side divisions, as they can simplify complex area comparisons.

In a right-angled triangle ABC with ∠B = 90°, the length of side AB = 8 units and the length of side BC = 6 units. Point D is on side AC such that AD: DC=2:3. Point E is on side BC such that BE: EC = 3: 4. If the area of triangle BDE is maximized, what is the ratio of the areas of triangles BDE and ABC?

Explore how to calculate the ratio of areas between triangles BDE and ABC in a right-angled triangle where sides are divided in specific ratios. This geometry problem applies the Pythagorean theorem, triangle area formulas, and segment division principles. Ideal for students in Grades 9-11.

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