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 7 - 2014 - Paper 1

In triangle ABC, we have:

$$BC = 7.5 cm,$$ $$CD = 6.1 cm.$$

To find angle DBC, we can use the Cosine Rule:

ext{cos}(DBC) = rac{AB^2 + BC^2 - CD^2}{2 imes AB imes BC}

We need to find AB. Since $EAB$ is a right triangle:

AB = 5 imes ext{sin}(EAB) = 5 imes ext{sin}igg(rac{i}{2}igg) = 5.

Substituting the known values:

ext{cos}(DBC) = rac{5^2 + 7.5^2 - 6.1^2}{2 imes 5 imes 7.5}

Calculating this gives:

ightarrow ext{angle DBC} = ext{acos(0.587)} ightarrow DBC hickapprox 0.943 ext{ radians}.$$$

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

ext{Area of triangle EAB} = rac{1}{2} imes base imes height = rac{1}{2} imes 5 imes 5 = 12.5 ext{ cm}^2.

Now, summing both areas:

$$$$

Thus, the area of the shape ABCDEA is approximately 30.0 cm².