Angle A of parallelogram ABCD α = 69.2°, AB = 3.58 and AD = 
5.46. Let's draw a perpendicular BE from B to side AD and from point E 
the perpendicular EF to the diagonal BD. Calculate EF.

To calculate \( EF \), which is the perpendicular from point \( E \) on side \( AD \) to the diagonal \( BD \), let's break down the problem into steps:

### Step 1: Understanding the Parallelogram
In parallelogram \( ABCD \):
- \( \alpha = 69.2^\circ \) (angle \( A \)),
- \( AB = 3.58 \) and \( AD = 5.46 \).

We are required to find the length of \( EF \), the perpendicular from point \( E \) on side \( AD \) to diagonal \( BD \).

### Step 2: Calculate the Length of \( BE \)
\( BE \) is the perpendicular dropped from \( B \) to side \( AD \). The length of \( BE \) can be calculated using trigonometry, specifically the sine of angle \( \alpha \):

\[
BE = AB \cdot \sin(\alpha)
\]

Substituting the values:

\[
BE = 3.58 \cdot \sin(69.2^\circ)
\]

Calculating \( \sin(69.2^\circ) \):

\[
\sin(69.2^\circ) \approx 0.9336
\]

\[
BE \approx 3.58 \cdot 0.9336 \approx 3.343
\]

So, \( BE \approx 3.34 \) units.

### Step 3: Calculate the Length of \( EF \)
Now, \( EF \) is the perpendicular from point \( E \) on side \( AD \) to diagonal \( BD \). The length of \( EF \) can be determined using the following relationship:

\[
EF = BE \cdot \cos(\alpha)
\]

Substituting the values:

\[
EF = 3.343 \cdot \cos(69.2^\circ)
\]

Calculating \( \cos(69.2^\circ) \):

\[
\cos(69.2^\circ) \approx 0.3584
\]

\[
EF \approx 3.343 \cdot 0.3584 \approx 1.198
\]

### Final Answer
The length \( EF \) is approximately \( 1.20 \) units.

Would you like more details or have any questions?

Here are 5 related questions:
1. How can you calculate the area of parallelogram ABCD using the given sides and angle?
2. What are the properties of the diagonals in a parallelogram?
3. How would the calculation change if a different angle was given?
4. What other methods can be used to find the height of a parallelogram?
5. How does the relationship between sine and cosine affect calculations in trigonometry?

**Tip:** When dealing with parallelograms, knowing how to decompose angles and sides into their perpendicular components is crucial for solving related geometric problems.

Learn how to calculate EF, the perpendicular from point E on side AD to diagonal BD in parallelogram ABCD with specific dimensions and angles. Detailed step-by-step solution involving trigonometric functions and parallelogram properties.

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