Let's solve the problem step by step for each diagram. The task is to calculate the perimeter of $ABCD$ in both diagrams, giving answers to two decimal places.

Diagram 1: Left Triangle $ABCD$

1. Given Data:

   • $AB = 12 \, \text{cm}$ (horizontal side),
   • $BD = 11 \, \text{cm}$ (diagonal side),
   • $\angle BDC = 30^\circ$.

2. Find missing sides:

   • Side $DC$: Use the right triangle $BDC$. $DC = BD \cdot \sin(30^\circ) = 11 \cdot 0.5 = 5.5 \, \text{cm}.$
   • Side $BC$: Use the right triangle $BDC$. $BC = BD \cdot \cos(30^\circ) = 11 \cdot \frac{\sqrt{3}}{2} \approx 11 \cdot 0.866 = 9.53 \, \text{cm}.$
   • Side $AD = AB - BC$: $AD = 12 - 9.53 = 2.47 \, \text{cm}.$

3. Perimeter of $ABCD$: Add up all the sides: $\text{Perimeter} = AB + BD + DC + AD = 12 + 11 + 5.5 + 2.47 = 30.97 \, \text{cm}.$

Diagram 2: Right Triangle $ABCD$

1. Given Data:

   • $AB = 3 \, \text{cm}$,
   • $BD = \sqrt{12} \approx 3.46 \, \text{cm}$,
   • $\angle BCD = 60^\circ$.

2. Find missing sides:

   • Side $DC$: Use the right triangle $BCD$. $DC = BD \cdot \sin(60^\circ) = 3.46 \cdot \frac{\sqrt{3}}{2} \approx 3.46 \cdot 0.866 = 3.00 \, \text{cm}.$
   • Side $BC$: Use the right triangle $BCD$. $BC = BD \cdot \cos(60^\circ) = 3.46 \cdot 0.5 = 1.73 \, \text{cm}.$
   • Side $AD = AB = 3 \, \text{cm}$ (as given, since $AB$ is equal to $AD$).

3. Perimeter of $ABCD$: Add up all the sides: $\text{Perimeter} = AB + BD + DC + BC = 3 + 3.46 + 3.00 + 1.73 = 11.19 \, \text{cm}.$

Final Answers:

• Perimeter of Diagram 1: $30.97 \, \text{cm}$,
• Perimeter of Diagram 2: $11.19 \, \text{cm}$.

Would you like a deeper explanation of any steps or formulas used?

Related Questions:

1. How do we derive the sine and cosine formulas for triangles?
2. Why is $\sin(30^\circ) = 0.5$ and $\cos(30^\circ) = \frac{\sqrt{3}}{2}$?
3. Can we use Pythagoras' theorem for solving right triangle sides instead of trigonometry here?
4. How would the calculation change if $\angle BDC$ were $45^\circ$?
5. What are some other practical applications of finding perimeters using trigonometry?

Tip:

Always check if the given triangle is a right triangle before deciding which formula (Pythagoras or trigonometry) is more efficient to use.