SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Measurement, STD1 M3 2025 HSC 20

A map of a park containing a duck pond is shown.

A fence is built passing through the points \(A\), \(B\) and \(C\) around the duck pond.
 

  1. Using the scale provided on the map, calculate the length of the fence \(AB\).   (2 marks)
  1. The length of \(AB\) is equal to the length of \(BC\).
  2. Use Pythagoras’ theorem to calculate the length of \(AC\) in metres. Give your answer correct to 3 significant figures.   (3 marks)
  3. What is the true bearing of point \(A\) from point \(C\) ?   (2 marks)
Show Answers Only

a.    \(75\ \text{metres}\)

b.    \(106\ \text{metres}\)

c.    \(225^\circ\)

Show Worked Solution

a.   \(\text{Scale: 1 grid width = 5 metres}\)

\(AB = 15 \times 5 = 75\ \text{metres}\)
 

b.   \(\text{Using Pythagoras:}\)

\(AC^2=AB^2+BC^2\)

\(AC^2=75^2+75^2=11250\)

  \(\therefore\ AC\) \(=\sqrt{11250}\)
    \(=106.066…\)
    \(\approx 106\ \text{m (3 sig fig)}\)

  
c.    \(\text{Since }AB=BC:\)

\(\angle BAC=\angle BCA=45^\circ\)

\(\text{Bearing of \(A\) from \(C\)}\ =180+45=225^\circ\)

Measurement, STD1 M3 2024 HSC 14

A hotel is located 186 m north and 50 m west of a train station.
 

  1. What is the straight line distance from the hotel to the train station? Round your answer to the nearest metre.   (2 marks)

  2. What is the bearing of the hotel from the train station? Round your answer to the nearest degree.   (2 marks)

Show Answers Only

a.    \(193\ \text{m}\)

b.    \(345^\circ\ \text{(nearest degree)}\)

Show Worked Solution

a.   \(\text{By Pythagoras:}\)

  \(d^2\) \(=50^2+186^2\)
  \(d^2\) \(=37\,096\)
  \(d\) \(=\sqrt{37\,096}\)
    \(=192.603\dots\)
    \(=193\ \text{m (nearest metre)}\)
♦♦ Mean mark (a) 37%.
b.     \(\tan\theta\) \(=\dfrac{50}{186}\)
  \(\theta\) \(=15.046\dots^\circ\)
    \(\approx 15^\circ\ \text{(nearest degree)}\)

 

\(\text{Bearing}\ H\ \text{from}\ T =360-15=345^\circ\)

♦♦♦ Mean mark (b) 12%.

Measurement, STD1 M3 2023 HSC 29

The diagram shows the location of three places \(X\), \(Y\) and \(C\).

\(Y\) is on a bearing of 120° and 15 km from \(X\).

\(C\) is 40 km from \(X\) and lies due west of \(Y\).

\(P\) lies on the line joining \(C\) and \(Y\) and is due south of \(X\).
  

  1. Find the distance from \(X\) to \(P\).  (2 marks)

  2. What is the bearing of \(C\) from \(X\), to the nearest degree?  (2 marks)

Show Answers Only
  1. \(7.5\ \text{km}\)
  2. \(259^{\circ}\)
Show Worked Solution

a.    \(\text{In}\ \Delta XPY:\)

\(\angle PXY=180-120=60^{\circ}\)

\(\cos 60^{\circ}\) \(=\dfrac{XP}{15}\)  
\(XP\) \(=15\times \cos 60^{\circ}\)  
  \(=7.5\ \text{km}\)  

♦♦ Mean mark (a) 24%.

b.    \(\text{In}\ \Delta XPC:\)

\(\text{Let}\ \theta = \angle CXP\)

\(\cos \theta\) \(=\dfrac{7.5}{40}\)  
\(\theta\) \(=\cos^{-1} \Big(\dfrac{7.5}{40}\Big)\)  
  \(=79.193…\)  
  \(=79^{\circ}\ \text{(nearest degree)}\)  

 

\(\text{Bearing}\ C\ \text{from}\ X\) \(=180+79\)  
  \(=259^{\circ}\)  

♦♦♦♦ Mean mark (b) 9%.

Measurement, STD2 M6 SM-Bank 4

The diagram shows three checkpoints A, B and C. Checkpoint C is due east of Checkpoint A. The bearing of Checkpoint B from Checkpoint A is N22°E and the bearing of Checkpoint C from Checkpoint B is S68°E. The distance between Checkpoint A and Checkpoint B is 42 kilometres.
 


 

  1. Mark the given information on the diagram and explain why `angleABC` is 90°.  (2 marks)

  2. Find the distance, to the nearest kilometre, between Checkpoint A and Checkpoint C.  (2 marks)

  3. If a runner is travelling 12.6 km/h, how long does it take her to travel between Checkpoint A and Checkpoint B, in hours and minutes?  (2 marks)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `112\ text{km}`
  3. `3text(h 20 mins)`
Show Worked Solution
i.   
`angle ABC` `= 22 + 68`
  `= 90°`

 

ii.  `text(In)\ \ DeltaABC,`

`cosangleBAC` `= (AB)/(AC)`
`cos68°` `= 42/(AC)`
`AC` `= 42/(cos68°)`
  `= 112.11…`
  `= 112\ text{km  (nearest km)}`

 

iii.    `text(Travel time)` `= text(dist)/text(speed)`
    `= 42/12.6`
    `= 3.333…`
    `= 3text(h 20 mins)`

Measurement, STD2 M6 SM-Bank 7 MC

Jeet walks 5 km from his home on a bearing of 153°. He then walks due north until he arrives a point which is due east of his home.

How far east, to the nearest 0.1 km, is Jeet from home?

  1.  2.3 km
  2.  2.5 km
  3.  4.9 km
  4.  9.8 km
Show Answers Only

`A`

Show Worked Solution

`text(Jeet finishes at)\ P`

`text(Find)\ \ OP:`

`cos 63°` `= (OP)/5`
`:. OP` `= 5 xx cos 63°`
  `= 2.26…`

 
`=> A`

Measurement, STD2 M6 2018 HSC 7 MC

The diagram shows the positions of towns `A`, `B` and `C`.

Town `A` is due north of town `B` and `angleCAB = 34°`
  


 

What is the bearing of town `C` from town `A`?

  1. 034°
  2. 146°
  3. 214°
  4. 326°
Show Answers Only

`C`

Show Worked Solution

`text(Bearing of Town)\ C\ text(from Town)\ A:`
 

`text(Bearing)` `= 180 + 34`
  `= 214^@`

 
`=>C`

Measurement, STD2 M6 2014 HSC 23 MC

The following information is given about the locations of three towns `X`, `Y` and `Z`: 

• `X` is due east of  `Z`

• `X` is on a bearing of  145°  from  `Y` 

• `Y` is on a bearing of  060°  from  `Z`. 

Which diagram best represents this information?
 

HSC 2014 23mci

Show Answers Only

`C`

Show Worked Solution
♦ Mean mark 38%
COMMENT: Drawing a parallel North/South line through `Y` makes this question much simpler to solve.

`text(S)text(ince)\ X\ text(is due east of)\ Z`

`=> text(Cannot be)\ B\ text(or)\ D`
 

 
`text(The diagram shows we can find)`

`/_ZYX = 60 + 35^@ = 95^@`

`text(Using alternate angles)\ (60^@)\ text(and)`

`text(the)\ 145^@\ text(bearing of)\ X\ text(from)\ Y`

`=>  C`

Measurement, STD2 M6 2010 HSC 10 MC

A plane flies on a bearing of  150° from  `A`  to  `B`.
 

Capture3

 
What is the bearing of  `A` from `B`?

  1. `30^@`
  2. `150^@`
  3. `210^@`
  4. `330^@`
Show Answers Only

`D`

Show Worked Solution
♦♦ Mean mark 34%

Capture3-i
 

`/_TBA=30^@\ \ \ text{(angle sum of triangle)}`

`:.\ text(Bearing of)\ A\ text{from}\ B`

`=360-30`

`=330^@`

`=>  D`