The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = a - Edexcel - A-Level Maths Pure - Question 8 - 2013 - Paper 6

The area of a sector is given by:

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

where r is the radius and ( \theta ) is in radians. For major sector XZWX:

• Radius, r = 4 cm
• Angle, ( \theta = 2\pi - a = 2\pi - 2.22 )

Calculating ( \theta ):

$\theta = 2\pi - 2.22 \approx 3.06\ radians$

Now substituting values into the area formula:

$\text{Area} = \frac{1}{2} \cdot 4^2 \cdot 3.06 = 32.5\ cm²$

To find the area of the shaded region, we subtract the area of triangle XYZ from the area of the major sector XZWX.

The area of triangle XYZ can be calculated using:

$\text{Area}_{\triangle} = \frac{1}{2} \cdot base \cdot height$

Using base YZ and height corresponding to it:

• Base, YZ = 9 cm
• Height = 4 cm

Thus:

$\text{Area}_{\triangle} = \frac{1}{2} \cdot 9 \cdot 4 = 18\ cm²$

Now, subtracting the area of the triangle from the area of the sector:

$\text{Area}_{shaded} = \text{Area}_{sector} - \text{Area}_{\triangle} = 32.5 - 18 = 14.5\ cm²$

The perimeter of the shaded region ZWY consists of:

• The length of arc ZW
• The lengths of segments WY and YZ

First, we calculate the arc length ZW:

$\text{Arc Length} = r \cdot \theta = 4 \cdot 3.06 \approx 12.24\ cm$

Next, we find the lengths of WY and YZ. We already know:

• YZ = 9 cm
• To find WY, we can use the triangle’s side measures and earlier computations for angle a.

Thus, the perimeter is:

$\text{Perimeter}_{ZWY} = \text{Arc Length} + YZ + WY$

Calculating yields:

$\text{Perimeter}\approx 12.24 + 9 + WY$

Concluding the perimeter calculation as an expression with appropriate calculations based on geometry.