Let's solve the problem based on the uploaded image.

(a) Work out the area of the parallelogram:

The formula for the area of a parallelogram is:
$\text{Area} = \text{Base} \times \text{Height}$
Alternatively, when the angle between two sides is given:
$\text{Area} = a \times b \times \sin(\theta)$
where:

• $a = 9.3 \, \text{cm}$ (side $EF$),
• $b = 14.7 \, \text{cm}$ (side $FG$),
• $\theta = 106^\circ$ (angle $EFG$).

First, calculate $\sin(106^\circ)$:
$\sin(106^\circ) \approx 0.961$.

Now substitute into the formula:
$\text{Area} = 9.3 \times 14.7 \times 0.961$
$\text{Area} \approx 131.2 \, \text{cm}^2$

Answer: The area of the parallelogram is 131 cm² (to 3 significant figures).

(b) Work out the length of diagonal $EG$:

The formula for the diagonal of a parallelogram is derived using the law of cosines:
$c^2 = a^2 + b^2 - 2ab\cos(\theta)$
where:

• $a = 9.3 \, \text{cm}$,
• $b = 14.7 \, \text{cm}$,
• $\theta = 106^\circ$.

First, calculate $\cos(106^\circ)$:
$\cos(106^\circ) \approx -0.2756$.

Now substitute into the formula:
$EG^2 = 9.3^2 + 14.7^2 - 2(9.3)(14.7)(-0.2756)$
$EG^2 = 86.49 + 216.09 + 75.595$
$EG^2 \approx 378.175$
$EG \approx \sqrt{378.175} \approx 19.4 \, \text{cm}$

Answer: The length of diagonal $EG$ is 19.4 cm (to 3 significant figures).

Would you like a step-by-step breakdown of any part of this solution?

Related Questions:

1. What is the formula for calculating the sine of an angle?
2. How does the law of cosines apply to non-right triangles?
3. Can we calculate the area using a different method (e.g., without trigonometry)?
4. What is the significance of rounding to 3 significant figures?
5. How does the cosine of an obtuse angle differ from an acute angle?

Tip: Always keep your calculator in degree mode unless working in radians!