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GEOMETRY, FUR1 2021 VCAA 4 MC

The side length of an equilateral triangle is 4 cm, as shown in the diagram below.
 

Which one of the following is not a correct calculation for the area of this triangle?

  1. `{sqrt3 xx 4^2}/{4}`
  2. `2 xx sqrt12`
  3. `1/2 xx 4 xx 4`
  4. `4^2/2 xx sin(60^@)`
  5. `sqrt{6 (6-4)^3}`
Show Answers Only

`C`

Show Worked Solution

`text{By Pythagoras:}`

`h = sqrt{4^2 -2^2} = sqrt12 = 2 sqrt3`
 

`text{Area}` `= 2 xx (1/2 xx 2 xx 2 sqrt3)`
  `= 4 sqrt3 \ text{cm}^2`

 
`text{Option C is the only option} ≠ 4 sqrt3`
 
`=> C`

GEOMETRY, FUR2 2012 VCAA 3

A tree is growing near the block of land.

The base of the tree, `T`, is at the same level as the corners, `P` and `S`, of the block of land.
 

Geometry and trigonometry, FUR2 2012 VCAA 3
 

  1. Show that, correct to two decimal places, distance `ST` is 41.81 metres.  (1 mark)
  2. From point `S`, the angle of elevation to the top of the tree is 22°.

     

    Calculate the height of the tree.

     

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

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `16.9\ text(m)`
Show Worked Solution
a.    `∠STP` `= 180 − (72 + 47)`
    `= 61^@`

 

 
`text(Using the sine rule:)`

`(ST)/(sin47^@)` `= 50/(sin61^@)`
`ST` `= (50 xx sin47^@)/(sin61^@)`
  `= 41.809…`
  `= 41.81\ text{m  (to 2 d.p.)  … as required}`

 

b.   `text(Let)\ \ X\ text(be the top of the tree)`

♦♦ Exact data unavailable although this part was highlighted as “very poorly answered”.

 
GEOMETRY, FUR2 2012 VCAA 3 Answer

`text(In)\ DeltaSXT, XT\ text(is height of tree.)`

`tan22^@` `= (XT)/(41.81)`
`:. XT` `= 41.81 xx tan22^@`
  `= 16.892…`
  `= 16.9\ text{m  (1 d.p.)}`

GEOMETRY, FUR2 2013 VCAA 1

A spectator, `S`, in the grandstand of an athletics ground is 13 m vertically above point `G`.

Competitor `X`, on the athletics ground, is at a horizontal distance of 40 m from `G`.
  

Geometry and Trig, FUR2 2013 VCAA 1_1
 

  1. Find the distance, `SX`, correct to the nearest metre.  (1 mark)

Competitor `X` is 40 m from `G` and competitor `Y` is 52 m from `G`.

The angle `XGY` is 113°.
 

Geometry and Trig, FUR2 2013 VCAA 1_2
 

    1. Calculate the distance, `XY`, correct to the nearest metre.  (1 mark)
    2. Find the area of triangle `XGY`, correct to the nearest square metre.  (1 mark)
  2. Determine the angle of elevation of spectator `S` from competitor `Y`, correct to the nearest degree.

    Note that `X`, `G` and `Y` are on the same horizontal level.  (1 mark)
     

Geometry and Trig, FUR2 2013 VCAA 1_3

Show Answers Only
  1. `42\ text{m  (nearest m)}`
    1. `77\ text{m  (nearest m)}`
    2. `957\ text{m²  (nearest m²)}`
  3. `14^@\ \ text{(nearest degree)}`
Show Worked Solution

a.   `text(Using Pythagoras:)`

`SX^2` `= 40^2 + 13^2`
  `= 1769`
`:. SX` `= 42.05…`
  `= 42\ text{m  (nearest m)}`

 

b.i.   `text(Using the cosine rule:)`

`XY^2` `= 40^2 + 52^2 – 52^2 xx 40 xx 52cos113^@`
  `= 5929.44…`
`:. XY` `= 77.00…`
  `= 77\ text{m  (nearest m)}`

 

b.ii.   `text(Using the sine rule:)`

`text(Area)` `= 1/2ab sinC`
  `= 1/2 xx 40 xx 52 xx sin113^@`
  `= 957.32…`
  `= 957\ text{m²  (nearest m²)}`

 

c.    Geometry-FUR2-2013-VCAA-1 Answer
`theta` `= text(angle of elevation of)\ S\ text(from)\ Y`
`tantheta` `= 13/52`
`:. theta` `= tan^(−1)\ 1/4`
  `= 14.036…`
  `= 14^@\ \ text{(nearest degree)}`

GEOMETRY, FUR2 2014 VCAA 2

The chicken coop has two spaces, one for nesting and one for eating.

The nesting and eating spaces are separated by a wall along the line `AX`, as shown in the diagrams below.
 

GEOMETRY, FUR2 2014 VCAA 21
 

`DX = 3.16\ text(m), ∠ADX = 45^@ and ∠AXD = 60^@.`

  1. Write down a calculation to show that the value of  `theta`  is 75°.  (1 mark)
  2. The sine rule can be used to calculate the length of the wall  `AX`.

     

    Fill in the missing numbers below.  (1 mark)
     

     

          GEOMETRY, FUR2 2014 VCAA 22
     

  3. What is the length of  `AX`?

     

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

  4. Calculate the area of the floor of the nesting space, `ADX`.

     

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

The height of the chicken coop is 1.8 m.

 

Wire mesh will cover the roof of the eating space.

 

The area of the walls along the lines  `AB, BC and CX`  will also be covered with wire mesh.

  1. What total area, in square metres, will be covered by wire mesh?

     

    Write your answer, correct to the nearest square metre.  (2 mark)

Show Answers Only
  1. `75^@`
  2. `(AX)/(sin 45^@) = (3.16)/(sin 75^@)`
  3. `2.31`
  4. `3.2\ text(m)^2`
  5. `17\ text(m)^2`
Show Worked Solution
a.    `theta` `= 180^@ − (45^@ + 60^@)`
    `= 75^@`

 

b.   `(AX)/(sin 45^@) = (3.16)/(sin 75^@)`

 

c.    `(AX)/(sin 45^@)` `= (3.16)/(sin 75^@)`
     
  `:. AX` `= (3.16 xx sin45^@)/(sin 75^@)`
    `= 2.313…`
    `=2.31\ text{m  (2 d.p.)}`

 

d.    `text(Area of)\ \ ΔADX` `= 1/2 xx b xx h`
    `= 1/2 xx 3.16 xx 2`
    `= 3.2\ text{m²  (1 d.p.)}`

 

e.   VCAA 2014 fur2 Q2ei
MARKER’S COMMENT: An excellent example where many students do not read the question carefully and give away easy marks.
`text(Roof area)` `=1/2 xx BC xx (AB + XC)`
  `= 1/2 xx 2 xx (3 + 1.84)`
  `= 4.84`

 

`text(Wall area)` `= (1.8 xx 3) + (1.8 xx 2) + (1.8 xx 1.84)`
  `= 12.312`

 

`:.\ text(Total area)` `= 4.84 + 12.312`
  `=17.152`
  `=17\ text{m²  (nearest m²)}`

GEOMETRY, FUR2 2015 VCAA 4

Wires support the communications tower, as shown in the diagram below.
 

Geometry and Trig, FUR2 2015 VCAA 4
 

The shortest wire is 31 m long.

The shortest wire makes an angle of 38° with the communications tower.

The longest wire is 37 m long.

The longest wire is attached to the communications tower `x` metres above the shortest wire.

What is the value of `x`?

Write your answer in metres, correct to one decimal place.  (2 marks)

Show Answers Only

`7.3\ text(m)`

Show Worked Solution

VCAA 2015 fur2 Q4bi

`text(Using the sine rule:)`

♦ Mean mark sub 50% (exact data not available).
`(sin theta)/31` `= sin142^@/37`
`sin theta` `= (31 xx sin142^@)/37`
`:. theta` `=31.05…^@`

 

`:.phi` `=180 – 142 – 31.05…`
  `= 6.94…^@`

 

`text(Using the cosine rule:)`

`x^2` `=37^2+31^2 – 2 xx 31 xx 37 xx cos 6.95^@`
  `=52.85…`
`:. x`  `=7.27…`
  `=7.3\ text(m)`

GEOMETRY, FUR1 2008 VCAA 5 MC

GEOMETRY, FUR1 2008 VCAA 5 MC
 

For the triangle shown, the value of  `sin x^@`  is given by

A.    `(sin 125.1^@)/2`

B.    `(5^2 + 4^2 − 8^2)/(2 xx 5 xx 4)`

C.    `2 xx sin 125.1^@`

D.   `(5^2 + 8^2 − 4^2)/(2 xx 5 xx 8)`

E.   `(5 xx sin 125.1^@)/8`

Show Answers Only

`A`

Show Worked Solution

`text(Using sine rule:)`

♦ Mean mark 41%.
`(sin x^@)/4` `= (sin 125.1^@)/8`
`:. sin x^@` `= (4 xx sin 125.1^@)/8`
  `= (sin 125.1^@)/2`

 
`=>  A`

GEOMETRY, FUR1 2006 VCAA 2 MC

GEOMETRY, FUR1 2006 VCAA 2 MC 

The length of  `RT`  in the triangle shown is closest to

A.   `17\ text(cm)`

B.   `33\ text(cm)`

C.   `45\ text(cm)`

D.   `53\ text(cm)`

E.   `57\ text(cm)`

Show Answers Only

`D`

Show Worked Solution

`text(Using sine rule:)`

`(RT)/sin108^@` `= 45/sin54^@`
 `RT` `= (45 xx sin108^@)/sin54^@`
  `= 52.90…`

 
`=>  D`

GEOMETRY, FUR1 2011 VCAA 4 MC

In triangle  `ACB`,  `/_ CAB = 60^@`  and  `/_ ABC = 80^@` 

The length of side  `AB = 50\text(m.)`
 


 

The length of side  `AC`  is closest to

A.    `57\ text(m)`

B.    `67\ text(m)`

C.    `77\ text(m)`

D.    `81\ text(m)`

E.   `100\ text(m)`

Show Answers Only

`C`

Show Worked Solution
`/_  ACB` ` = 180 – (60 + 80) \ \ text{(angle sum of}\ Delta ABC text{)}`
  ` = 40 ^@`

 

`text(Using sine rule:)`

`(AC) / (sin 80^@)` `= 50/(sin 40^@)`
`:. AC`  `= (50 xx sin 80^@) / (sin 40^@)`
  `= 76.60…\ \ text(m)`

 
`=> C`

GEOMETRY, FUR1 2012 VCAA 7 MC

`PQR` is a triangle with side lengths `x, 10` and `y`, as shown below.

In this triangle, angle `RPQ = 37°` and angle `QRP = 42°.`

Which one of the following expressions is correct for triangle `PQR`?

A.    `x = 10/(sin 37°)`

B.    `y = 10/(tan 37°)`

C.    `x = 10 × (sin 42°)/(sin 37°)`

D.    `y = 10 × (sin 37°)/(sin 101°)`

E.    `10^2 = x^2 + y^2 - 2xy cos 42°`

Show Answers Only

`C`

Show Worked Solution

`∠ PQR` `= 180 – (37 + 42)\ \ \ \ text {(angle sum of}\ ΔPQR text{)}`
  `= 101°`

 

`text (Using the sine rule:)`

`y/sin 101= x/sin 42= 10/sin 37`

`:. x= 10 × (sin 42)/(sin 37)`

 
`rArr C`

GEOMETRY, FUR1 2010 VCAA 3 MC

An equilateral triangle of side length 6 cm is cut from a sheet of cardboard.

A circle is then cut out of the triangle, leaving a hole of diameter 2 cm as shown below.
 

The area of cardboard remaining, as shown by the shaded region in the diagram above, is closest to

A.    `3\ text(cm²)`

B.    `9\ text(cm²)`

C.  `12\ text(cm²)`

D.  `15\ text(cm²)`

E.  `16\ text(cm²)`

Show Answers Only

`C`

Show Worked Solution

`text(Equilateral triangle)\ =>\ 3 xx 60°\ text(angles.)`

`text(Area of)\ Delta` `= 1/2 ab sin C`
  `= 1/2 xx 6 xx 6 xx sin 60°`
  `= 15.58\ text(cm²)`

 

`text(Area of circle)` `= pir^2`
  `= pi xx 1^2`
  `= 3.14\ text(cm²)`

 
`:.\ text(Area of cardboard remaining)`

`= 15.58- 3.14`

`= 12.44\ text(cm²)`

 
`=> C`