Figure 1 shows the triangle ABC, with AB = 8 cm, AC = 11 cm and ∠ BAC = 0.7 radians - Edexcel - A-Level Maths Pure - Question 8 - 2005 - Paper 2

To find the perimeter of region R, we need to calculate the length of side BC first. We can use the cosine rule:

$$BC^2 = AB^2 + AC^2 - 2 \times AB \times AC \times \cos(\theta)$$

Substituting the values, we have:

$$BC^2 = 8^2 + 11^2 - 2 \times 8 \times 11 \times \cos(0.7).$$

After calculating, we find:

$$BC \approx 7.098 \text{ cm}.$$

Now, we can find the perimeter:

$$\text{Perimeter} = AB + AC + BC = 8 + 11 + 7.098 \approx 26.098 \text{ cm}.$$

Rounded to three significant figures, the perimeter is approximately 26.1 cm.

To find the area of region R, we need to calculate the area of triangle ABC and the area of sector ADB separately.

1. Area of Triangle ABC: Using the formula:

$$\text{Area} = \frac{1}{2} ab \sin(\theta)$$

where $a = 11$ cm, $b = 8$ cm, and $\theta = 0.7$ radians, we compute:

$$\text{Area} = \frac{1}{2} \times 11 \times 8 \times \sin(0.7) \approx 28.3 \text{ cm}^2.$$

2. Area of Sector ADB: Using the formula for the area of a sector:

$$\text{Area of sector} = \frac{1}{2} r^2 \theta$$

For radius $r = 8$ cm:

$$\text{Area of sector} = \frac{1}{2} \times 8^2 \times 0.7 \approx 22.4 \text{ cm}^2.$$

Finally, we find the area of region R:

$$\text{Area of R} = \text{Area of triangle} - \text{Area of sector} \approx 28.3 - 22.4 \approx 5.9 \text{ cm}^2.$$

Rounded to three significant figures, the area of region R is approximately 5.95 cm².