To find the length of the perpendicular $EF$ from point $E$ (where $BE$ is perpendicular to $AD$) to the diagonal $BD$, we will use the following steps:

Step-by-Step Solution:

1. Calculate the length of BE (Perpendicular from B to AD):

   In parallelogram $ABCD$, $\angle A = 69.2^\circ$, $AB = 3.58$, and $AD = 5.46$.

   Since $BE$ is perpendicular to $AD$, we can use the sine of angle $\alpha = 69.2^\circ$ to find $BE$:

   $\sin(\alpha) = \frac{BE}{AB}$

   Solving for $BE$:

   $BE = AB \cdot \sin(\alpha) = 3.58 \cdot \sin(69.2^\circ)$

2. Calculate the length of diagonal BD:

   In a parallelogram, the length of diagonal $BD$ can be calculated using the law of cosines in $\triangle ABD$:

   $BD^2 = AB^2 + AD^2 - 2 \cdot AB \cdot AD \cdot \cos(\alpha)$

   Substituting the known values:

   $BD^2 = 3.58^2 + 5.46^2 - 2 \cdot 3.58 \cdot 5.46 \cdot \cos(69.2^\circ)$

3. Find the distance EF (Perpendicular from E to diagonal BD):

   Now, we need to find the length of $EF$, which is the perpendicular distance from point $E$ to diagonal $BD$.

   To calculate $EF$, we use the area of $\triangle BDE$ in two different ways: one using base $BE$ and height $EF$, and the other using base $BD$ and height $BE$. Equating both expressions for the area, we can solve for $EF$:

   The area of $\triangle BDE$ can be given by:

   $\text{Area} = \frac{1}{2} \times BE \times EF = \frac{1}{2} \times BD \times BE$

Let's compute these values step by step.Here are the computed values:

1. Length of $BE$: $3.35$ units (perpendicular from $B$ to $AD$)
2. Length of $BD$: $5.36$ units (diagonal of the parallelogram)
3. Length of $EF$: $3.13$ units (perpendicular from $E$ to diagonal $BD$)

So, the length of the perpendicular $EF$ is approximately 3.13 units.

Would you like more details, or do you have any questions?

Here are some related questions that could help you deepen your understanding:

1. How do you calculate the area of a parallelogram using its sides and angles?
2. What are the properties of diagonals in a parallelogram?
3. How can trigonometry be used to solve problems involving non-right triangles?
4. Why is the law of cosines useful for finding diagonal lengths in a parallelogram?
5. How would you find the length of another diagonal of the parallelogram?

Tip: Always double-check the angle measures and side lengths to ensure accurate calculations, especially in geometric problems.