Let's work through the problem step by step based on the diagram and the questions provided:

Given:

• Triangle ABC with a perpendicular line BD from B to AC.
• AB = 5.5 m, BD = 4 m, DC = 5 m.

(a) Calculate the length of BC using Pythagoras' theorem

In right triangle BDC: $BC^2 = BD^2 + DC^2$

Substitute the values: $BC^2 = 4^2 + 5^2 = 16 + 25 = 41$

$BC = \sqrt{41} \approx 6.4 \, \text{m}.$

(b) Calculate the length of the side AD using Pythagoras' theorem

In right triangle ABD: $AB^2 = AD^2 + BD^2$

Substitute the values: $5.5^2 = AD^2 + 4^2$

$30.25 = AD^2 + 16$

$AD^2 = 30.25 - 16 = 14.25$

$AD = \sqrt{14.25} \approx 3.77 \, \text{m}.$

(c) Calculate the angles $\angle BCA$, $\angle CBD$, $\angle ABD$, and $\angle BAD$

1. $\angle CBD$: Using $\tan$ in triangle BDC: $\tan(\angle CBD) = \frac{BD}{DC} = \frac{4}{5}$

   $\angle CBD = \tan^{-1}\left(\frac{4}{5}\right) \approx 38.66^\circ$

2. $\angle BCA$: Since $\triangle ABC$ is a right triangle: $\angle BCA = 90^\circ - \angle CBD \approx 90^\circ - 38.66^\circ = 51.34^\circ$

3. $\angle ABD$: Using $\tan$ in triangle ABD: $\tan(\angle ABD) = \frac{BD}{AD} = \frac{4}{3.77}$

   $\angle ABD = \tan^{-1}\left(\frac{4}{3.77}\right) \approx 46.62^\circ$

4. $\angle BAD$: Again, since $\triangle ABD$ is a right triangle: $\angle BAD = 90^\circ - \angle ABD \approx 90^\circ - 46.62^\circ = 43.38^\circ$

Summary of Results

• (a) $BC \approx 6.4 \, \text{m}$
• (b) $AD \approx 3.77 \, \text{m}$
• (c) Angles:
  • $\angle BCA \approx 51.34^\circ$
  • $\angle CBD \approx 38.66^\circ$
  • $\angle ABD \approx 46.62^\circ$
  • $\angle BAD \approx 43.38^\circ$

Do you want further explanations on any part?
Here are 5 additional questions to expand on the concepts:

1. How can Pythagoras' theorem be generalized for 3D space?
2. What is the significance of trigonometric ratios in solving roof problems?
3. Can you find the area of triangle ABC?
4. How would the problem change if BD were not perpendicular to AC?
5. How do these calculations apply in real-world engineering projects?

Tip: Always confirm the perpendicularity of sides when using trigonometry or Pythagoras' theorem!