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, e) the perimeter ZWY 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 7 - 2013 - Paper 6

Question 7

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 7 - 2013 - Paper 6

Step 1

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

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Answer

To find angle a, we use the cosine rule:

$c^2 = a^2 + b^2 - 2ab imes ext{cos}(C)$

Let XZ = 4 cm, YZ = 9 cm, and XY = 6 cm, we need to find cos(a).

Substituting these values in:

$9^2 = 4^2 + 6^2 - 2 imes 4 imes 6 imes ext{cos}(a)$

Calculating gives:

$81 = 16 + 36 - 48 imes ext{cos}(a)$

This simplifies to:

$81 = 52 - 48 imes ext{cos}(a)$

Rearranging gives:

$48 imes ext{cos}(a) = -29$

Thus:

$ext{cos}(a) = - rac{29}{48}$

Using a calculator yields:

a ≈ 2.22 radians.

Step 2

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

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Answer

The area of a sector is given by the formula:

$ext{Area} = rac{1}{2} r^2 heta$

For the major sector XZW:

Radius, r = 4 cm
Angle, θ = 2π - 2.22 radians

Calculating the area:

$ext{Area} = rac{1}{2} imes 4^2 imes (2 ext{π} - 2.22)$

Computing yields:

$= 32.5 ext{ cm}^2$

Step 3

the area of this shaded region.

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Answer

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

Area of triangle XYZ: Using the formula:

$ext{Area} = rac{1}{2} imes b imes h$

Base YZ = 9 cm and height from X to YZ can be derived from angle a: height = 4 × sin(2.22).

Calculating gives: Area of triangle XYZ ≈ 12.96 cm².

Area of shaded region:

$ext{Area(shaded)} = ext{Area(sector)} - ext{Area(triangle)}$

So: $32.5 - 12.96 = 19.54 ext{ cm}^2$

Step 4

the perimeter ZWY of this shaded region.

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Answer

The perimeter of shaded region ZWY includes the arc length ZW and lines WY and YZ.

Arc length ZW:
- From radius and angle:
- $ext{Length} = r imes θ = 4 imes 2.22 = 8.88 ext{ cm}$
Perimeter:
- $ext{Perimeter} = ZY + WY + ext{Arc ZW}$
- We already know ZY = 9 cm
- WY is derived from triangle XYZ;

Total perimeter is approximately:

$= 9 + WY + 8.88$

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 7 - 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 7 - 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 ZWY of this shaded region.

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