The diagram shows triangle `ABC`.
Calculate the area of the triangle, to the nearest square metre. (3 marks)
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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`
`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` |
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.
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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` |
`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.)}` |
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.)}`
`text(Find)\ AC:`
`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.)}` |
The area of the triangle shown is 250 cm².
What is the value of `x`, correct to the nearest whole number?
`D`
`text(Using)\ \ \ A = 1/2ab\ sin\ C`
`250` | `= 1/2 xx 30x\ sin\ 44^@` |
`= 15x\ sin\ 44 ^@` | |
`:.x` | `= 250/(15\ sin\ 44^@)` |
`= 23.99…\ text(m)` |
`=>D`
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.
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i. `text(Box height) = 15\ text(cm)`
`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`
`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)}` |
What is the area of this triangle, to the nearest square metre?
`C`
`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`