Consider a triangle ABC such that B is a right angle. Let D and E be points on AC such that the triangle BDE is equilateral. If the area of triangle BDE is half of the area of triangle ABC, calculate the ratio of AC to DE.

We are tasked with finding the ratio of \( AC \) to \( DE \) given that triangle \( ABC \) is a right triangle with \( \angle B = 90^\circ \), and triangle \( BDE \) is equilateral. Additionally, the area of triangle \( BDE \) is half of the area of triangle \( ABC \).

### Step 1: Notation and Setup
Let:
- \( AB = c \),
- \( BC = b \),
- \( AC = a \).

The area of triangle \( ABC \) is:
\[
\text{Area of } \triangle ABC = \frac{1}{2} \cdot AB \cdot BC = \frac{1}{2} \cdot c \cdot b.
\]

Let \( s \) denote the side length of the equilateral triangle \( BDE \), i.e., \( BD = BE = DE = s \). The area of an equilateral triangle with side length \( s \) is given by:
\[
\text{Area of } \triangle BDE = \frac{\sqrt{3}}{4} s^2.
\]

### Step 2: Use the Given Condition on Areas
We are told that the area of triangle \( BDE \) is half of the area of triangle \( ABC \), i.e.,
\[
\frac{\sqrt{3}}{4} s^2 = \frac{1}{2} \cdot \frac{1}{2} \cdot c \cdot b.
\]
Simplifying this equation:
\[
\frac{\sqrt{3}}{4} s^2 = \frac{1}{4} \cdot c \cdot b.
\]
Multiplying both sides by 4:
\[
\sqrt{3} s^2 = c \cdot b.
\]
Thus, the side length \( s \) of the equilateral triangle \( BDE \) is:
\[
s^2 = \frac{c \cdot b}{\sqrt{3}},
\]
\[
s = \sqrt{\frac{c \cdot b}{\sqrt{3}}}.
\]

### Step 3: Ratio of \( AC \) to \( DE \)
Now, we need to calculate the ratio \( \frac{AC}{DE} \). Since \( AC = a \) and \( DE = s \), the ratio is:
\[
\frac{AC}{DE} = \frac{a}{s}.
\]
We already know that \( s = \sqrt{\frac{c \cdot b}{\sqrt{3}}} \), so the ratio becomes:
\[
\frac{AC}{DE} = \frac{a}{\sqrt{\frac{c \cdot b}{\sqrt{3}}}}.
\]

Finally, this expression simplifies to give the desired ratio of \( AC \) to \( DE \).

Would you like further details on this simplification or have any questions?

Here are 5 related questions for further exploration:
1. How do the side lengths \( AB \) and \( BC \) affect the side length of \( DE \)?
2. What is the significance of the ratio \( \frac{AC}{DE} \) in other triangle configurations?
3. How can we derive the formula for the area of an equilateral triangle?
4. What happens if the area of \( BDE \) were a different fraction of \( ABC \)'s area?
5. How would the problem change if triangle \( ABC \) were not a right triangle?

**Tip:** The relationship between area and side lengths often provides a path to discovering ratios in geometric problems.

Explore the geometric problem of calculating the ratio of AC to DE in a right triangle where BDE forms an equilateral triangle. This problem includes key concepts like triangle area formulas and properties of equilateral triangles. Suitable for high school students (Grades 9-11).

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