The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = a. The point W lies on the line XY. The circular arc ZW, in Figure 1 is a major arc of the circle with centre X and radius 4 cm. (a) Show that, to 3 significant figures, a = 2.22 radians. (b) Find the area, in cm², of the major sector XZW. (c) The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown shaded in Figure 1. Calculate d) the area of this shaded region, d) the perimeter ZYWZ of this shaded region.

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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

Question 8

The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = a. The point W lies on the line XY. The circular arc ZW, in Figure 1 is a major arc... show full transcript

Worked Solution & Example Answer: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

Step 1

Show that, to 3 significant figures, a = 2.22 radians.

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Answer

To find the angle a, we can use the cosine rule in triangle XYZ:

XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(a)

Substituting the values we have:

4^2 = 6^2 + 9^2 - 2 \cdot 6 \cdot 9 \cdot \cos(a)

This simplifies to:

16 = 36 + 81 - 108 \cdot \cos(a)

Therefore:

108 \cdot \cos(a) = 117 \Rightarrow \cos(a) = \frac{117 - 16}{108} = \frac{101}{108}

Calculating this gives:

a \approx 2.22 \text{ radians (to 3 significant figures)}

Step 2

Find the area, in cm², of the major sector XZW.

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Answer

The formula for the area of a sector is:

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

Substituting the radius (r = 4 cm) and angle (\theta = 2.22 radians):

\text{Area} = \frac{1}{2} \cdot 4^2 \cdot 2.22 = 16 \cdot 1.11 = 32.5 \, ext{cm}^2

Step 3

the area of this shaded region.

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Answer

To find the area of the shaded region, subtract the area of triangle XYZ from the area of the sector XZW:

Area of triangle XYZ is given by:

\text{Area} = \frac{1}{2} \cdot 4 \cdot 6 \cdot \sin(2.22)

Calculating, we find the area of triangle XYZ:

\text{Area} \approx 12 \cdot 0.78 \approx 9.36 \, ext{cm}^2

Thus, the area of the shaded region is:

\text{Shaded Area} = 32.5 - 9.36 \approx 23.14 \, ext{cm}^2

Step 4

the perimeter ZYWZ of this shaded region.

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Answer

To find the perimeter ZYWZ of the shaded region, we calculate:

Arc length ZW:
$\text{Arc Length} = r \theta = 4 \cdot 2.22 = 8.88 \, ext{cm}$
The lengths WY and YZ are:
$WY = 9 \, ext{cm}, \, YZ = 6 \, ext{cm}$
Therefore, the total perimeter is:
$\text{Perimeter} = ZW + WY + YZ = 8.88 + 9 + 6 = 23.88 \, ext{cm}$

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

Worked Solution & Example Answer: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

Show that, to 3 significant figures, a = 2.22 radians.

Find the area, in cm², of the major sector XZW.

the area of this shaded region.

the perimeter ZYWZ of this shaded region.

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