Let's examine and solve each part of the provided problem.

Problem 1b

The problem describes a rectangle $ABCD$ with two equilateral triangles $EBF$ and $AGD$ inside it. We are given:

• $BE = 4e \sqrt{3}$
• We need to show that the area $A$ of rectangle $ABCD$ is $36e^2 \sqrt{3}$ without using rounded values.

To approach this, we would analyze the properties of the equilateral triangles $EBF$ and $AGD$ and use them to determine the side lengths of the rectangle $ABCD$, eventually leading to the area.

1. Since $EBF$ and $AGD$ are equilateral, all their sides are equal.
2. The length $BE = 4e \sqrt{3}$ suggests that the height of the triangle $EBF$ (using the standard height formula for an equilateral triangle) could be a basis for calculating the length of the sides of the rectangle.
3. Using the information and properties of equilateral triangles, we can derive that each side of the rectangle is related to $e$.

Problem 2a

Here, we are given a right triangle $ABC$ and an isosceles triangle $ADE$ with the following details:

• $AB = 13.2$ cm
• $\alpha = 55^\circ$
• $CE = 8.0$ cm
• $AE = DE$

The tasks are:

1. Calculate the length of $DF$.
2. Calculate the perimeter of quadrilateral $ABFE$.

To solve this, we proceed as follows:

1. Length of $DF$:

   • Using trigonometric relationships, we can use $\angle \alpha$ and known side lengths in triangle $ABC$ to find missing distances, including $DF$.

2. Perimeter of $ABFE$:

   • Once $DF$ and other necessary distances are calculated, we can sum up the lengths of $AB$, $BF$, $FE$, and $EA$ to get the perimeter.

Would you like detailed calculations for each of these steps?

Additional Questions to Consider:

1. How would the problem change if $BE$ were given in a different form?
2. What trigonometric identities are most useful in right triangles?
3. How can the properties of equilateral triangles simplify calculations in geometry?
4. Why is it significant that $AE = DE$ in an isosceles triangle?
5. How does the relationship between the angles in the triangle affect the calculations?

Tip:

In geometry problems involving equilateral and right triangles, remember that symmetry often allows simplifications, especially using height and area formulas directly tied to side lengths.