smc-804-30-Sine Rule (Area)

Measurement, STD2 M6 2024 HSC 32

A regular pentagon \(ABCDE\) is drawn inside a circle with a radius 30 cm.

\(O\) is the centre of the circle.

What is the area of the shaded region of the circle. Answer correct to 2 significant figures. (4 marks)

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\(\text{690 cm}^{2}\)

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\(\text{Method 1}\)

\(\text{Area}\ \Delta ODC\)	\(= \dfrac{1}{2} ab \sin C\)
	\(=\dfrac{1}{2} \times 30 \times 30 \times \sin 72^{\circ} \)
	\(=427.98\ \text{cm}^{2}\)

\(\text{Shaded Area}\)	\(=\ \text{Area of circle}-5 \times\ \text{Area}\ \Delta ODC\)
	\(=(\pi \times 30^2)-(5 \times 427.98)\)
	\(=687.5\)
	\(=690\ \text{cm}^{2}\ \text{(2 sig.fig)}\)

\(\text{Method 2}\)

\(\text{Consider}\ \Delta ODC:\)

\(\text{Area}=427.98\ \text{cm}^{2}\ \ \text{(see above)}\)

\(\text{Area of sector}\ ODC = \dfrac{72}{360} \times \pi \times 30^{2} = 565.49\ \text{cm}^{2}\)

\(\text{Shaded area (total)}\)	\(=(565.49-427.98) \times 5\)
	\(=690\ \text{cm}^{2}\ \text{(2 sig.fig)}\)

Measurement, STD2 M6 2023 HSC 35

The diagram shows triangle `ABC`.

Calculate the area of the triangle, to the nearest square metre. (3 marks)

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`147\ text{m}^2`

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`text{Using the sine rule:}`

`(CB)/sin60^@`	`=12/sin25^@`
`CB`	`=sin60^@ xx 12/sin25^@`
	`=24.590…`

`angleACB=180-(60+25)=95^@\ \ text{(180° in Δ)}`

`text{Using the sine rule (Area):}`

`A`	`=1/2 xx AC xx CB xx sin angleACB`
	`=1/2 xx 12 xx 24.59 xx sin95^@`
	`=146.98…`
	`=147\ text{m}^2`

Measurement, STD2 M6 2021 HSC 32

A right-angled triangle `XYZ` is cut out from a semicircle with centre `O`. The length of the diameter `XZ` is 16 cm and `angle YXZ` = 30°, as shown on the diagram.

Find the length of `XY` in centimetres, correct to two decimal places. (2 marks)

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Hence, find the area of the shaded region in square centimetres, correct to one decimal place. (3 marks)

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`13.86 \ text{cm}`
`45.1 \ text{cm}^2`

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a.	`cos 30^@`	`=(XY)/16`
	`XY`	`= 16 \ cos 30^@`
		`= 13.8564`
		`= 13.86 \ text{cm (2 d.p.)}`

b.	`text{Area of semi-circle}`	`= 1/2 times pi r^2`
		`= 1/2 pi times 8^2`
		`= 100.531 \ text{cm}^2`

♦ Mean mark part (b) 36%.

`text{Area of} \ Δ XYZ`	`= 1/2 ab\ sin C`
	`= 1/2 xx 16 xx 13.856 xx sin 30^@`
	`= 55.42 \ text{cm}^2`

`:. \ text{Shaded Area}`	`= 100.531-55.42`
	`= 45.111`
	`= 45.1 \ text{cm}^2 \ \ text{(1 d.p.)}`

Measurement, STD2 M6 2018 HSC 30c

The diagram shows two triangles.

Triangle `ABC` is right-angled, with `AB = 13 text(cm)` and `/_ABC = 62°`.

In triangle `ACD, \ AD = x\ text(cm)` and `/_DAC = 40°`. The area of triangle `ACD` is 30 cm².

What is the value of `x`, correct to one decimal place? (3 marks)

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`8.1\ text{cm (1 d.p.)}`

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`text(Find)\ AC:`

♦ Mean mark 39%.

`sin62°`	`= (AC)/13`
`AC`	`= 13 xx sin62°`
	`= 11.478…`

`text(Using the sine rule in)\ DeltaACD :`

`text(Area)`	`= 1/2 xx AC xx AD xx sin40°`
`30`	`= 1/2 xx 11.478… xx x xx sin40°`
`:.x`	`= (30 xx 2)/(11.478… xx sin40°)`
	`= 8.13…`
	`= 8.1\ text{cm (1 d.p.)}`

Measurement, STD2 M6 2015 HSC 22 MC

The area of the triangle shown is 250 cm².

What is the value of `x`, correct to the nearest whole number?

`11`
`18`
`22`
`24`

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`D`

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`text(Using)\ \ \ A = 1/2ab\ sin\ C`

♦ Mean mark 42%.

`250`	`= 1/2 xx 30x\ sin\ 44^@`
`250`	`= 15x\ sin\ 44 ^@`
`:.x`	`= 250/(15\ sin\ 44^@)`
	`= 23.99…\ text(m)`

`=>D`

Measurement, STD2 M6 2006 HSC 9 MC

What is the area of this triangle, to the nearest square metre?

`text(152 m²)`
`text(283 m²)`
`text(328 m²)`
`text(351 m²)`

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`C`

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`text(Using the Sine rule)`

`A`	`= 1/2 ab\ sin C`
	`= 1/2 xx 39 xx 47 xx sin 21^@`
	`=\ text(328.44… m²)`

`=> C`

Measurement, STD2 M6 2004 HSC 9 MC

What is the area of the triangle to the nearest square metre?

`text(102 m²)`
`text(153 m²)`
`text(172 m²)`
`text(178 m²)`

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`C`

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`text(Using sine rule,)`

`text(Area)`	`= 1/2 ab sin C`
	`= 1/2 xx 30 xx 20 xx sin 35^@`
	`=172.072…\ text(m²)`

`=> C`

Measurement, STD2 M6 2008 HSC 25c

Pieces of cheese are cut from cylindrical blocks with dimensions as shown.

Twelve pieces are packed in a rectangular box. There are three rows with four pieces of cheese in each row. The curved surface is face down with the pieces touching as shown.

What are the dimensions of the rectangular box? (4 marks)

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To save packing space, the curved section is removed.
What is the volume of the remaining triangular prism of cheese? Answer to the nearest cubic centimetre. (2 marks)

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`41\ text(cm) xx 21\ text(cm) xx 15\ text(cm)`
`506\ text(cm)³\ text{(nearest whole)}`

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i. `text(Box height) = 15\ text(cm)`

♦ Mean mark 45%.

`text{(radius of the arc)}`

`text(Box width)`	`= 3 xx 7`
	`= 21\ text(cm)`
`text(Box length)`	`= 4x`

`text(Using cosine rule)`

`c^2`	`= a^2 + b^2 – 2ab cos C`
`x^2`	`= 15^2 + 15^2 – 2 xx 15 xx 15 xx cos 40^@`
	`= 450 – 344.7199…`
	`= 105.2800…`
`x`	`= 10.2606…`

`text(Box length)`	`= 4 xx 10.2606…`
	`= 41.04…`

`:.\ text(Dimensions are)\ \ 41\ text(cm) xx 21\ text(cm) xx 15\ text(cm)`

ii. `text(Volume) = Ah`

♦♦♦ Mean mark 22%.

`h = 7\ text(cm)`

`text(Find)\ A:`

`A`	`= 1/2 ab sin C`
	`= 1/2 xx 15 xx 15 xx sin 40^@`
	`= 72.3136…`

`:. V`	`= 72.3136… xx 7`
	`= 506.195…`
	`= 506\ text(cm³)\ \ text{(nearest whole)}`

Measurement, STD2 M6 2012 HSC 10 MC

What is the area of this triangle, to the nearest square metre?

`33\ text(m²)`
`37\ text(m²)`
`42\ text(m²)`
`44\ text(m²)`

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`C`

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`text(Let unknown angle)=/_C`

`/_C`	`= 180-(50 + 57)\ \ \ \ \ (180^@ \ text(in)\ Delta)`
	`=73^@`

`:. A`	`= 1/2 ab\ sinC`
	`= 1/2 xx 9.9 xx 8.8 xx sin73^@`
	`= 41.656 \ text(m²)`

`=> C`