The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B - Edexcel - A-Level Maths Pure - Question 5 - 2014 - Paper 1

Since the points A, B, and C are collinear, we can find the angle DBC using the relationship of the angles in triangle DBC.

We know:

• Angle EBD = 1.4 radians
• Angle EAB = ( \frac{\pi}{2} ) radians.

Thus:

$\text{Angle DBC} = \text{Angle EAB} - \text{Angle EBD} = \frac{\pi}{2} - 1.4$

Calculating:

$\text{Angle DBC} \approx 3.14159/2 - 1.4 \approx 0.943 \text{ radians}$

Therefore, the size of angle DBC is approximately 0.943 radians.

To find the total area of the shape ABCDEA, we need to sum the area of triangle EAB and the area of sector BDE.

Step 1: Area of Triangle EAB

The area of triangle EAB can be calculated using:

$\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$

Here:

• The base AE is 5 cm.
• The height BE is 6.1 cm.

Thus:

$\text{Area} = \frac{1}{2} \times 5 \times 6.1 = 15.25 \text{ cm}^2$

Step 2: Total Area

Now summing the areas:

$\text{Total Area ABCDEA} = \text{Area of Triangle EAB} + \text{Area of Sector BDE}$

$\text{Total Area} = 15.25 + 17.5 = 32.75 \text{ cm}^2$

Rounding to 3 significant figures, the area of shape ABCDEA is 32.8 cm².