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GEOMETRY, FUR1 2020 VCAA 3 MC

Two trees stand on horizontal ground.

A 25 m cable connects the two trees at point A and point B, as shown in the diagram below.
 


 

Point A is 45 m above the ground and point B is 30 m above the ground.

The horizontal distance, in metres, between point A and point B is

  1. 10
  2. 15
  3. 20
  4. 30
  5. 35
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`C`

Show Worked Solution

`CB` `= sqrt(AB^2 – AC^2)`
  `= sqrt(25^2 – 15^2)`
  `= 20`

 
`=>  C`

GEOMETRY, FUR1 2017 VCAA 2 MC

A right-angled triangle, `XYZ`, has side lengths  `XY = 38.5\ text(cm)`  and  `YZ = 24.0\ text(cm)`, as shown in the diagram below.
 


 

The length of  `XZ`, in centimetres, is closest to

  1. `24.8`
  2. `30.1`
  3. `38.8`
  4. `45.4`
  5. `62.5`
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`D`

Show Worked Solution

`text(By Pythagoras:)`

`XZ` `= sqrt(38.5^2 + 24^2)`
  `= sqrt(2058.25)`
  `= 45.36…\ text(cm)`

 
`=> D`

GEOMETRY, FUR2 2016 VCAA 2

Salena practises golf at a driving range by hitting golf balls from point  `T`.

The first ball that Salena hits travels directly north, landing at point  `A`.

The second ball that Salena hits travels 50 m on a bearing of 030°, landing at point  `B`.

The diagram below shows the positions of the two balls after they have landed.
  

  1. How far apart, in metres, are the two golf balls?  (1 mark)
  2. A fence is positioned at the end of the driving range.

     

    The fence is 16.8 m high and is 200 m from the point  `T`.


     
     
    What is the angle of elevation from  `T`  to the top of the fence?

     

    Round your answer to the nearest degree.  (1 mark) 

Show Answers Only
  1. `25\ text(m)`
  2. `5^@\ text{(nearest degree)}`
Show Worked Solution

a.   `text(Let)\ \ d\ text(= distance apart)`

`sin30^@` `= d/50`
`:. d` `= 50 xx sin 30^@`
  `= 25\ text(m)`

 

b.   `text(Let)\ \ x^@\ text(= angle of elevation from)\ T`

`tanx` `= 16.8/200`
  `= 0.084`
`:. x` `= 4.801…`
  `= 5^@\ text{(nearest degree)}`

GEOMETRY, FUR2 2006 VCAA 2

An allotment of land contains a communications tower, `PQ`.

Points `S`, `Q` and `T` are situated on level ground.

From `S` the angle of elevation of `P` is 20°.

Distance `SQ` is 125 metres.

Distance `TQ` is 98 metres. 
 

Geometry and Trig, FUR2 2006 VCAA 2
 

  1. Determine the height, `PQ`, of the communications tower.

     

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

  2. Determine the angle of depression of `T` from `P`.

     

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

Show Answers Only
  1. `45.5\ text{m (1 d.p.)}`
  2. `24.9^@\ text{(1 d.p.)}`
Show Worked Solution

a.   `text(In)\ Delta PQS,`

♦ Mean mark of both parts (combined) was 50%.
`tan 20^@` `= (PQ)/125`
`:. PQ` `= 125 xx tan 20^@`
  `= 45.49…`
  `= 45.5\ text{m (1 d.p.)}`

 

b.     

 

`theta = text(angle of depression of)\ T\ text(from)\ P`

`tan theta` `= 45.5/98`
  `= 0.464…`
`:. theta` `= 24.90…`
  `= 24.9^@\ text{(1 d.p.)}`

 

GEOMETRY, FUR2 2009 VCAA 1

A ferry, `F`, is 400 metres from point `O` at the base of a 50 metre high cliff, `OC`.
 

GEOMETRY, FUR2 2009 VCAA 1 
 

  1. Show that the gradient of the line `FC` in the diagram is 0.125.  (1 mark)
  2. Calculate the angle of elevation of point `C` from `F`.

     

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

  3. Calculate the distance `FC`, in metres, correct to one decimal place.  (1 mark)
Show Answers Only
  1. `text(See Worked Solution.)`
  2. `7.1^@`
  3. `403.1\ text(m)`
Show Worked Solution
a.    `text(Gradient)` `= text(rise)/text(run)`
    `= 50/400`
    `= 0.125\ \ text(…   as required)`

 

b.    `tan\ /_ CFO` `= 50/400`
  `:. /_ CFO` `= tan^-1 0.125`
    `= 7.12…`
    `= 7.1^@\ text{(1 d.p.)}`

 

c.    `text(Using Pythagoras:)`
  `FC` `= sqrt(400^2 + 50^2)`
    `= 403.11…`
    `= 403.1\ text(m)`

GEOMETRY, FUR1 2007 VCAA 2 MC

GEOMETRY, FUR1 2007 VCAA 2 MC
 

For an observer on the ground at `A`, the angle of elevation of a weather balloon at `B` is 37°.

`C` is a point on the ground directly under the balloon. The distance `AC` is 2200 m.

To the nearest metre, the height of the weather balloon above the ground is

A.   `1324\ text(m)`

B.   `1658\ text(m)`

C.   `1757\ text(m)`

D.   `2919\ text(m)`

E.   `3655\ text(m)`

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`B`

Show Worked Solution
`tan 37^@` `= (BC)/2200`
`:. BC`  `= 2200 xx tan 37^@`
  `= 1657.8…\ text(m)`

 
`=>  B`

GEOMETRY, FUR1 2011 VCAA 2 MC


 

The point `Q` on building `B` is visible from the point `P` on building `A`, as shown in the diagram above.

Building `A` is 16 metres taller than building `B`.

The horizontal distance between point `P` and point `Q` is 23 metres.

The angle of depression of point `Q` from point `P` is closest to

A.    `35°`

B.    `41°`

C.    `44°`

D.    `46°`

E.    `55°`

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`A`

Show Worked Solution

`tan theta` `= 16/23`
`theta` `= 34.8^@`

 

`:.\ text(Angle of depression from)\ P\ text(to)\ Q = 34.8^@`

`text{(alternate,}\ PS\ text(||)\ RQ text{)}`

`=> A`

GEOMETRY, FUR1 2014 VCAA 1 MC

The top of a ladder that is 4.50 m long rests 3.25 m up a wall, as shown in the diagram below.
 

The angle, `theta`, that the ladder makes with the wall is closest to

A.   `36°`

B.   `44°`

C.   `46°`

D.   `50°`

E.   `54°`

Show Answers Only

`B`

Show Worked Solution
`cos theta` `=text(adj)/text(hyp)`
  `=3.25/4.50`
  `=0.722…` 
`∴ theta` `= 43.76°` 

 
`=>B`