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 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.

Learn how to calculate the length of perpendicular EF in parallelogram ABCD using trigonometric and geometric principles. This problem involves applying the law of cosines and trigonometric functions to find the length of EF. Suitable for high school students studying geometry and trigonometry.

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