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GEOMETRY, FUR1 2021 VCAA 3 MC

A photograph was enlarged by an area scale factor of 9.

The length of the original photograph was 12 cm.

The original photograph and the enlarged photograph are similar in shape.

The length of the enlarged photograph, in centimetres, is

  1.    4
  2.    9
  3.  27
  4.  36
  5. 108
Show Answers Only

`D`

Show Worked Solution

`text{Area scale factor} = 9`

♦ Mean mark 38%.

`text{Length sale factor} = sqrt9 = 3`

`:. \ text{Length of large photo}` `= 12 xx 3`
  `= 36 \ text{cm}`

`=> D`

GEOMETRY, FUR1 2020 VCAA 9 MC

Shot-put is an athletics field event in which competitors throw a heavy spherical ball (a shot) as far as they can.

The size of the shot for men and the shot for women is different.

The diameter of the shot for men is 1.25 times larger than the diameter of the shot for women.

The ratio of the total surface area of the women’s shot to the total surface area of the men’s shot is

  1.   1:4
  2.   1:25
  3.   4:5
  4.   5:4
  5. 16:25
Show Answers Only

`E`

Show Worked Solution

♦♦ Mean mark 36%.
`SA_f` `: SA_m` 
`4pir^2` `: 4 pi(5/4r)^2`
`1` `: 25/16`
`16` `: 25`

 
`=>  E`

GEOMETRY, FUR1-NHT 2019 VCAA 8 MC

A young tree is protected by a tree guard in the shape of a square-based pyramid.

The height of the tree guard is 54 cm, as shown in the diagram below.
 
 

         
 

The top section of the tree guard is removed along the dotted line to allow the tree to grow.

Removing this top section decreases the height of the tree guard to 45 cm, as shown in the diagram below.
 
 

         
 

The ratio of the volume of the tree guard that is removed to the volume of the tree guard that remains is

  1.  `1 : 6`
  2.  `1 : 35`
  3.  `1 : 36`
  4.  `1 : 215`
  5.  `1 : 216`
Show Answers Only

`D`

Show Worked Solution

`text(Top section is similar to full tree guard.)`

`text(Ratio of sides)` `= (54 – 45) : 54`
  `= 9:54`
  `= 1:6`

 

`:. \ text(V)_1 : text(V)_2` `= 1^3 : 6^3`
  `= 1 : 216`

 

`text(However V)_2 \ text(is the volume of a full tree guard. )`

`text(We need volume of the remaining tree guard.)`

`:. \ V_1 : V_2 \ text{(remains)} = 1:215`
 
`=> \ D`

GEOMETRY, FUR1-NHT 2019 VCAA 6 MC

A cake in the shape of three cylindrical sections is shown in the diagram below.
 

       
 

Each section of the cake has a height of 8 cm, as shown in the diagram.

The middle section of the cake, B, has twice the volume of the top section of the cake, A.

The bottom section of the cake, C, has twice the volume of the middle section of the cake, B.

The volume of the top section of the cake, A, is 900 cm3.

The diameter of the bottom section of the cake, C, in centimetres, is closest to

  1. 12
  2. 18
  3. 24
  4. 36
  5. 48
Show Answers Only

`C`

Show Worked Solution
`text(Volume) \ C` `= 900 xx 4`
  `= 3600 \ text(cm)^3`

 

`pi xx r_C^2 xx h` `= 3600`
`r_C^2` `= (3600)/(8 pi)`
`r_C^2` `= 11.968 \ …`

 
`:. \ text(Diameter of)\  C ≈ 24\ text(cm)`
  
`=> \ C`

GEOMETRY, FUR2 2017 VCAA 1

Miki is planning a gap year in Japan.

She will store some of her belongings in a small storage box while she is away.

This small storage box is in the shape of a rectangular prism.

The diagram below shows that the dimensions of the small storage box are 40 cm × 19 cm × 32 cm.
 

The lid of the small storage box is labelled on the diagram above.

    1. What is the surface area of the lid, in square centimetres?  (1 mark)
    2. What is the total outside surface area of this storage box, including the lid and base, in square centimetres?  (1 mark)
  2. Miki has a large storage box that is also a rectangular prism.

      

    The large storage box and the small storage box are similar in shape.

      

    The volume of the large storage box is eight times the volume of the small storage box.

      

    The length of the small storage box is 40 cm.

    What is the length of the large storage box, in centimetres?  (1 mark)

Show Answers Only

a.i.   `760\ text(cm²)`

a.ii.  `5296\ text(cm²)`

b.     `80\ text(cm)`

Show Worked Solution
a.i.    `text{Area (lid)}` `= 40 xx 19`
    `= 760\ text(cm²)`

 

a.ii.    `text(Total S.A.)` `= 2 xx (32 xx 19) + 2 xx (40 xx 32) + 2 xx 760`
    `= 5296\ text(cm²)`

 

b.   `text(If volume scale factor = 8,)`

♦♦ Mean mark 32%.
MARKER’S COMMENT: A lack of understanding between linear and volume scale factors was again notable.

`text(⇒ Linear scale factor) = root(3)8 = 2`

`:.\ text(Length of large storage box)`

`= 2 xx 40`

`= 80\ text(cm)`

GEOMETRY, FUR1 2016 VCAA 8 MC

A string of seven flags consisting of equilateral triangles in two sizes is hanging at the end of a racetrack, as shown in the diagram below.
 


 

The edge length of each black flag is twice the edge length of each white flag.

For this string of seven flags, the total area of the black flags would be

  1. two times the total area of the white flags.
  2. four times the total area of the white flags.
  3. `4/3` times the total area of the white flags.
  4. `16/3` times the total area of the white flags.
  5. `16/9` times the total area of the white flags. 
Show Answers Only

`D`

Show Worked Solution

`text(Shapes are similar.)`

♦♦♦ Mean mark 30%.

`text(Scale factor of sides = 2)`

`text(Scale factor of areas) = 2^2 = 4`

 

`text(S)text(ince there are 4 black and 3)`

`text(white flags,)`

`:.\ text{Total area of black flags (in white flags)}`

`= 4/3 xx 4`

`= 16/3 xx text(Area of white flags)`

`=> D`

GEOMETRY, FUR2 2012 VCAA 4

`OABCD` has three triangular sections, as shown in the diagram below. 

Triangle `OAB` is a right-angled triangle. 

Length `OB` is 10 m and length `OC` is 14 m. 

Angle `AOB` = angle `BOC` = angle `COD` = 30°
 

Geometry and trigonometry, FUR2 2012 VCAA 4
 

  1. Calculate the length, `OA`.

     

    Write your answer, in metres, correct to two decimal places.  (1 mark)

  2. Determine the area of triangle `OAB`.

     

    Write your answer, in m², correct to one decimal place.  (1 mark)

  3. Triangles `OBC` and `OCD` are similar.

     

    The area of triangle `OBC` is 35 m².

     

    Find the area of triangle `OCD`, in m².  (2 marks)

  4. Determine angle `CDO`.

     

    Write your answer, correct to the nearest degree.  (2 marks)

Show Answers Only
  1. `8.66\ text{m  (2 d.p.)}`
  2. `21.7\ text{m²  (1 d.p.)}`
  3. `68.6\ text(m²)`
  4. `43^@\ \ text{(nearest degree)}`
Show Worked Solution

a.   `text(In)\ DeltaOAB,`

`cos30^@` `= (OA)/10`
`:. OA` `= 10 xx cos30`
  `= 8.660…`
  `= 8.66\ text{m  (2 d.p.)}`

 

b.   `text(Using the sine rule,)`

`text(Area)\ DeltaAOB` `= 1/2 ab sinC`
  `= 1/2 xx 8.66 xx 10 xx sin30^@`
  `= 21.65…`
  `= 21.7\ text{m²  (1 d.p.)}`

 

c.   `text(Linear scale factor) = 14/10 = 1.4`

`:. (text(Area)\ DeltaOCD)/(text(Area)\ DeltaOBC)` `= 1.4^2`
   
`:. text(Area)\ DeltaOCD` `= 35 xx 1.4^2`
  `= 68.6\ text(m²)`

 

d.   `text(In)\ DeltaBCO,`

`BC^2` `= 10^2 + 14^2 – 2 xx 10 xx 14 xx cos30^@`
  `= 53.51…`
`:. BC` `= 7.315…\ text(m)`

 

`text(Using the sine rule in)\ DeltaBCO,`

`(sin angleBCO)/10` `= (sin30^@)/(BC)`
`sin angleBCO` `= (10 xx sin30)/(7.315…)`
  `= 0.683…`
`:. angleBCO` `= 43.11^@`

 

`text(S)text(ince)\ DeltaBCO\ text(|||)\ DeltaCDO,`

`angleCDO` `= angleBCO`
  `= 43^@\ \ text{(nearest degree)}`

GEOMETRY, FUR2 2014 VCAA 3

The chicken coop contains a circular water dish.

Water flows into the dish from a water container.

The water container is in the shape of a cylinder with a hemispherical top.

The water container and the dish are shown in the diagrams below.
 

GEOMETRY, FUR2 2014 VCAA 3
 

The cylindrical part of the water container has a diameter of 10 cm and a height of 15 cm.

The hemisphere has a radius of 5 cm.

  1. What is the surface area of the hemispherical top of the water container?

     

    Write your answer, correct to the nearest square centimetre.  (1 mark)

  2. What is the maximum volume of water that the water container can hold?

     

    Write your answer, correct to the nearest cubic centimetre.  (2 marks)

The eating space of the chicken coop also has a feed container.

 

The feed container is similar in shape to the water container.

 

The volume of the water container is three-quarters of the volume of the feed container.

 

The surface area of the water container is 628 cm².

  1. What is the surface area of the feed container?

     

    Write your answer, correct to the nearest square centimetre.  (2 marks) 

Show Answers Only
  1. `157\ text(cm)²`
  2. `1440\ text(cm)³`
  3. `761\ text(cm)²`
Show Worked Solution
a.    `text(Area)` `= 1/2 xx (4pi xx r^2)`
    `= 1/2 xx (4pi xx 5^2)`
    `= 157\ text{cm²  (nearest cm²)}`

 

b.    `text(Volume of Cylinder)` `=pi r^2 h`
    `=pi xx 5^2 xx 15`
    `=1178.09…\ text(cm³)`
  `text(Volume of Hemisphere)` `=1/2 xx 4/3 pi r^3`
    `=1/2 xx 4/3 xx pi xx 5^3`
    `=261.79…\ text(cm³)`

 

♦ Mean mark of all parts of question (combined) was 44%.

`:.\ text(Maximum volume of the container)`

`=1178.09… + 261.79…`

`=1439.8…`

`=1440\ text{cm³  (nearest cm³)}`

 

c.    `text(Volume Factor)` `=4/3`
  `text(Linear Factor)` `=root3(4/3)`
  `text(Area Factor)` `=(root3(4/3))^2=1.2114…`

 

`:.\ text(S.A. of Feed Container)`

`=628 xx 1.2114…`

`=760.7…`

`=761\ text(cm²)`

GEOMETRY, FUR2 2013 VCAA 4

Competitors in the intermediate division of the discus use a smaller discus than the one used in the senior division, but of a similar shape. The total surface area of each discus is given below.
 

GEOMETRY, FUR2 2013 VCAA 4
 

By what value can the volume of the intermediate discus be multiplied to give the volume of the senior discus?  (2 marks)

Show Answers Only

`1.728`

Show Worked Solution

`k^2 = 720/500 = 36/25`

♦♦ Mean mark well below 50% (exact data unavailable.
MARKER’S COMMENT: A common error was to use a ratio where `k<1` which was obviously wrong since the volume needed to be scaled up.
`:. k` `= sqrt(36/25) = 1.2`
`:. k^3` `= 1.2^3 = 1.728`

 

`:.\ text(Volume of intermediate discuss can)`

 `text(be multiplied by 1.728)`

GEOMETRY, FUR2 2007 VCAA 4

Tessa has a task that involves removing the top 24 cm of the height of her right regular pyramid below.

geo-2007-4 

The shape remaining is shown in Figure 5 below. The top surface, `JKLM`, is parallel to the base, `ABCD`.
 

GEOMETRY, FUR2 2007 VCAA 4
 

  1. What fraction of the height of the pyramid has Tessa removed to produce Figure 5?  (1 mark)
  2. What fraction of the volume of the pyramid remains in Figure 5?  (2 marks)
Show Answers Only
  1. `3/4`
  2. `37/64`
Show Worked Solution

a.   `text(Fraction of the height removed)`

`= 24/32`

`= 3/4`

 

b.   `text(Linear scale factor)\ =3/4`

`text(Volume scale factor) = (3/4)^3 = 27/64`

♦♦ Mean mark of both parts (combined) was 18%.

 

`:.\ text(Fraction of pyramid remaining)`

`= 1 – 27/64`

`= 37/64`

GEOMETRY, FUR2 2009 VCAA 4

The ferry has two fuel filters, `A` and `B`.

Filter `A` has a hemispherical base with radius 12 cm.

A cylinder of height 30 cm sits on top of this base.
 

GEOMETRY, FUR2 2009 VCAA 41
 

  1. Calculate the volume of filter `A`. Write your answer correct to the nearest cm³.  (2 marks)

Filter `B` is a right cone with height 50 cm.

 

GEOMETRY, FUR2 2009 VCAA 42

  1. Originally filter `B` was full of oil, but some was removed.

     

    If the height of the oil in the cone is now 20 cm, what percentage of the original volume of oil was removed?  (2 marks)

Show Answers Only
  1. `17\ 191\ text(cm³)`
  2. `text(93.6%)`
Show Worked Solution

a.   `text(Volume of cylinder)`

♦ Mean mark of all parts (combined) 36%.

`= pi r^2 h`

`= pi xx 12^2 xx 30`

`= 13\ 571.68…\ text(cm³)`
 

`text(Volume of hemisphere)`

`= 1/2 xx 4/3 pi r^3`

`= 1/2 xx 4/3 xx pi xx 12^3`

`= 3619.11…\ text(cm³)`

 

`:.\ text(Volume of filter)\ A`

`= 13\ 571.68… + 3619.11…`

`= 17\ 190.79…`

`= 17\ 191\ text(cm³)`

 

b.   `text(Linear Factor) = 20/50 = 2/5`

`:.\ text(Volume Factor) = (2/5)^3 = 8/125`

`:.\ text(Percentage of original volume removed)`

`= (1 – 8/125) xx 100 text(%)`

`= 93.6text(%)`

GEOMETRY, FUR1 2011 VCAA 9 MC

In the diagram below, `/_ ABD = /_ ACB = theta^@`.

`BD = 24\ text(cm)` and `BC = 40\ text(cm.)`

The area of triangle `ABD` is `100\ text(cm².)`
 

The area of triangle `ABC`, in cm² is closest to

A.  `167`

B.  `178`

C.  `267`

D.  `278`

E.  `378`

Show Answers Only

`D`

Show Worked Solution

`/_ ABD = /_ BCA = theta^@\ \ \ text{(given)}`

`/_ BAD\ text(is common)`

`:. Delta BAD\ text(|||)\ Delta CAB\ \ text{(equiangular)}`

 

`text(Linear scale factor)`

`k= (BC)/(DB) = 40/24 = 5/3`

`:.\ text(Area)\ Delta CAB` `= k^2 xx text(Area)\ Delta BAD`
  `= (5/3)^2 xx 100`
  `= 277.77…\ \ text(cm²)`

 
`=> D`

GEOMETRY, FUR1 2015 VCAA 9 MC

A wedge of cheese is in the shape of a triangular prism.

The base of the wedge of cheese is 8 cm long, as shown below.
 

GEO & TRIG, FUR1 2015 VCAA 9 MC

A smaller, similar wedge of cheese is cut from the larger wedge of cheese, as shown in the diagram.

The cut is made at a distance of `d` cm from the back edge of the larger wedge.

The volume of the smaller wedge is half the volume of the larger wedge.

The value of `d`, in centimetres, is closest to

A.   `1.7`

B.   `2.3`

C.   `4.0`

D.   `5.7`

E.   `6.3`

Show Answers Only

`B`

Show Worked Solution

`text(The right-angled triangle with the dotted line)`

`text(is similar to the large right-angled triangle.)`

`V\ text{(Large wedge)}` `= A_(text{large}) xx h`
`V\ text{(Small wedge)}` `= A_(text{small}) xx h`

 
`:.\ (A_(text{large}))/(A_(text{small}))=2`
 

`text(Linear factor)\ = 8 : (8-d)`
 

 `(8/(8 – d))^2` `=2`
`8/(8 – d)`  `= sqrt2`
`8-d` `=8/sqrt2`
`d` `=8-8/sqrt2`
  `=2.34…`

 
`=> B`

 

`text(The word “similar” was used in the stem of this question)`

`text(in its everyday sense. However, some students interpreted)`

`text(this word geometrically, so option)\ A\ text(was also accepted.)`

GEOMETRY, FUR1 2007 VCAA 5 MC

A block of land has an area of 4000 m².

When represented on a map, this block of land has an area of 10 cm².

On the map 1 cm would represent an actual distance of

A.       `10\ text(m)`

B.       `20\ text(m)`

C.       `40\ text(m)`

D.     `400\ text(m)`

E.   `4000\ text(m)`

Show Answers Only

`B`

Show Worked Solution

`text(Area scale factor) = k^2`

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: A majority of students obtained the the “area” scale factor (k² = 400) but failed to convert this to the corresponding linear scale factor.
`k^2` `= 4000/10`
  `= 400`
`:. k` `= sqrt(400)`
  `= 20`

 
`:.\ text(Linear scale factor) = 20,`

`text(i.e.)\ \ 1\ text(cm):20\ text(m)`

`=>  B`

GEOMETRY, FUR1 2013 VCAA 4 MC

A cafe sells two sizes of cupcakes with a similar shape.

The large cupcake is 6 cm wide at the base and the small cupcake is 4 cm wide at the base.
 

geometry20
 

The price of a cupcake is proportional to its volume.

If the large cupcake costs $5.40, then the small cupcake will cost

A.   `$1.60`

B.   `$2.32`

C.   `$2.40`

D.   `$3.40`

E.   `$3.60`

Show Answers Only

`A`

Show Worked Solution

`text(Linear scale factor of cupcake lengths)`

♦♦ Mean mark 22%.

`k = 4/6 = 2/3`
 

`text(Volume scale factor is)`

`k^3 = (2/3)^3=8/27`
 

`text(S)text(ince price is proportional to volume,)`

`text(Cost)_text(small)/text(Cost)_text(large)` `=8/27`
`:. \ text(Cost)_text(small)` `=(8 xx 5.40)/27`
  `=$1.60`

`=> A`

GEOMETRY, FUR1 2010 VCAA 4 MC

Cube `A` and cube `B` are shown below.

The side length of cube `A` is 1.5 times the side length of cube `B`.

The surface area of cube `B` is 256 cm².
 


 

The surface area of Cube `A` is

A.  `114\ text(cm²)`

B.  `256\ text(cm²)`

C.  `384\ text(cm²)`

D.  `576\ text(cm²)`

E.  `864\ text(cm²)`

Show Answers Only

`D`

Show Worked Solution

`text(Linear scale factor)`

♦ Mean mark 46%.

`k=1.5`

`:.\ text(S.A. Cube)\ A` `= k^2 xx text(S.A. Cube)\ B`
  `= 1.5^2 xx 256`
  `=576`

`=> D`