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Aussie Maths & Science Teachers: Save your time with SmarterEd

Similarity, SMB-028

The floor plan of a home unit has been drawn to scale.
 


 

What is the total floor area of the home unit in square metres?  (2 marks)

Show Answers Only

`text{55.5 m}^2`

Show Worked Solution

`text{Conversion: 1000 mm = 1 metre}`

`text{Calculate area by splitting into 2 rectangles}`

`text{Area}`  `=  5.5 × 5 + (4.5 +2.5) xx 4`  
   `=  55.5\ text{m}^2`  

Similarity, SMB-027

 


 

  1. What scale factor is used to convert Circle A into Circle B.   (1 mark)

  2. Complete this equation:
  3.      Area of Circle A = _____ × Area of Circle B.  (1 mark)

Show Answers Only
  1. \(\dfrac{1}{3}\)
  2. \(9\)
Show Worked Solution

i.     \(\text{Scale factor}\ = \dfrac{\text{Diameter B}}{\text{Diameter A}} = \dfrac{0.8}{2.4} = \dfrac{1}{3} \)
 

ii.     \(\text{Scale factor (B to A)} = \dfrac{\text{Diameter A}}{\text{Diameter B}} = \dfrac{2.4}{0.8} = 3 \)

\(\text{Scale factor (Area)} = 3^2 = 9 \)

\(\therefore\ \text{Area of Circle A = 9 × Area of Circle B} \)

Similarity, SMB-026

The two triangles below are similar.
 

Find the length of \(ED\).   (3 marks)

Show Answers Only

\(ED = 15\)

Show Worked Solution

\(\text{Using Pythagoras in}\ \Delta ABC: \)

\( AB=\sqrt{13^2-12^2}=\sqrt{25}=5 \)

\(\text{Scale factor}\ = \dfrac{FD}{AC} = \dfrac{39}{13} = 3 \)

\(\therefore ED = 3 \times AB = 3 \times 5 = 15 \)

Similarity, SMB-025

A triangular prism is pictured below.
 

By what factor will its volume change if

  1. Each dimension is doubled?   (1 mark)

  2. Each dimension is decreased by two-thirds?   (2 marks)
Show Answers Only
  1. \(\text{Increases by a factor of 8}\)
  2. \(\text{Decreases by a factor of}\ \ \dfrac{1}{27} \)
Show Worked Solution

i.    \(\text{Dimensions increase by a factor of 2}\)

\(\Rightarrow\ \text{Volume increases by a factor of}\ 2^3 = 8\)
 

ii.    \(\text{Dimensions decrease by two-thirds}\)

\(\Rightarrow\ \text{i.e. adjust dimensions by a factor of}\ \ \dfrac{1}{3} \)

\(\Rightarrow\ \text{Volume decreases by a factor of}\ \Big{(} \dfrac{1}{3} \Big{)}^3 = \dfrac{1}{27} \)

Similarity, SMB-024

Prove that the two triangles in the right cone pictured below are similar.   (2 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\angle ACD \ \text{is common to}\ \Delta ACE\ \text{and}\ \Delta BCD \)

\(\angle CAE=\angle CBD=90^\circ \ \text{(Right cone)} \)

\(\therefore \Delta ACE\ \text{|||}\ \Delta BCD\ \ \text{(equiangular)}\)

Similarity, SMB-023

Show that \(\Delta ACD\) and \(\Delta DCB\) in the figure below are similar.   (2 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\angle BDC = 180-(88+44) = 48^{\circ} \ \text{(angle sum of Δ)} \)

\(\angle CAD = \angle CDB = 48^{\circ} \)

\(\angle ACD=44^{\circ} \ \text{is common to}\ \Delta ACD\ \text{and}\ \Delta BCD \)

\(\therefore \Delta ACD\ \text{|||}\ \Delta DCB\ \ \text{(equiangular)}\)

Similarity, SMB-022

Prove that this pair of triangles are similar.   (2 marks)

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\angle PQT = \angle SQR \ \text{(vertically opposite)} \)

\(\dfrac{PQ}{QS} = \dfrac{2.5}{5} = \dfrac{1}{2} \)

\(\dfrac{TQ}{QR} = \dfrac{2}{4} = \dfrac{1}{2} \)

\(\therefore \Delta PQT\ \text{|||}\ \Delta SQR\ \ \text{(sides adjacent to equal angles in proportion)}\)

Similarity, SMB-021

Prove that this pair of triangles are similar.   (2 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\angle ACD = \angle BEA \ \text{(given)} \)

\(\angle BAE\ \text{is common to}\ \Delta ACD\ \text{and}\ \Delta BEA \)

\(\therefore \Delta ACD\ \text{|||}\ \Delta BEA\ \ \text{(equiangular)}\)

Similarity, SMB-020

Prove that this pair of triangles are similar.   (3 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\text{Find unknown side}\ (x)\ \text{of smaller triangle}\)

\(\text{Using Pythagoras:}\)

\(x=\sqrt{5^2-4^2} = \sqrt{9} = 3\)
 

\(\dfrac{AC}{DE} = \dfrac{5}{15} = \dfrac{1}{3} \)

\(\dfrac{BC}{EF} = \dfrac{3}{9} = \dfrac{1}{3} \)

\(\angle ABC = \angle DEF = 90^{\circ} \ \ \text{(given)} \)

\(\therefore \Delta ABC\ \text{|||}\ \Delta DEF\ \ \text{(hypotenuse and second side of right-angled triangle in proportion)}\)

Similarity, SMB-019

Prove that this pair of triangles are similar.   (2 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\dfrac{AB}{GH} = \dfrac{11}{22} = \dfrac{1}{2} \)

\(\dfrac{BC}{FG} = \dfrac{3.5}{7} = \dfrac{1}{2} \)

\(\angle ABC = \angle FGH = 95^{\circ} \ \ \text{(given)} \)
 

\(\therefore \Delta ABC\ \text{|||}\ \Delta HGF\ \ \text{(sides adjacent to equal angles in proportion)}\)

Similarity, SMB-018

Prove that this pair of triangles are similar.   (2 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\angle ABE = \angle CBD\ \ \text{(vertically opposite)} \)

\(\angle AEB = \angle BCD\ \ \text{(alternate,}\ AE \parallel CD \text{)} \)

\(\therefore \Delta ABE\ \text{|||}\ \Delta DBC\ \ \text{(equiangular)}\)

Similarity, SMB-017

Prove that this pair of triangles are similar.   (2 marks)
 

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution

\(\dfrac{RP}{AC} = \dfrac{18}{12} = \dfrac{3}{2} \)

\(\dfrac{QR}{BA} = \dfrac{12}{8} = \dfrac{3}{2} \)

\(\dfrac{PQ}{CB} = \dfrac{9}{6} = \dfrac{3}{2} \)
 

\(\therefore \Delta PQR\ \text{|||}\ \Delta CBA\ \ \text{(three pairs of sides in proportion)}\)

Similarity, SMB-016

Triangle I and Triangle II are similar. Pairs of equal angles are shown.
 

Find the area of Triangle II?  (3 marks)

Show Answers Only

`24\ text{cm}^2`

Show Worked Solution

`text(In Triangle I, using Pythagoras:)`

`text{Base}` `= sqrt(5^2-3^2)`
  `= 4`

 
`text(Triangle I ||| Triangle II (given))`

♦♦ Mean mark 29%.

`=>\ text(corresponding sides are in the same ratio)`

`text{Scale factor}\ = 6/2=2`

`text{Scale factor (Area)}\ = 2^2=4`

`:. text(Area (Triangle II))` `= 4 xx text{Area of triangle I}`
  `= 4 xx 1/2 xx 3 xx 4`
  `=24\ text{cm}^2`

Similarity, SMB-015

Towns `A, B` and `C` are marked on the scale diagram below.
  

The distance from Town `A` to Town `B` is 9 km.

What is the distance, in kilometres, from Town `B` to Town `C`?   (2 marks)

Show Answers Only

`3.6\ text(km)`

Show Worked Solution

`text(Let)\ \ d = text(distance from Town)\ B\ text(to Town)\ \ C`

`text(S)text(ince the diagram is to scale,)`

`d/9` `= 2/5`
`:. d` `= (9 xx 2)/5`
  `= 3.6\ text(km)`

Similarity, SMB-014 MC

Spiro is making a scale drawing of his house.

  • The height of the garage in the scale model is 6 centimetres.
  • The height of Spiro's actual garage is 3 metres.

What does 1 centimetre in Spiro's scale drawing represent in his real house?

  1. 1800 cm
  2. 200 cm
  3. 100 cm
  4. 50 cm
Show Answers Only

`D`

Show Worked Solution
`text(6 cm)` `= 3\ text(m)`
  `= 300\ text(cm)`
`:. 1\ text(cm)` `= 300 -: 6`
  `= 50\ text(cm)`

 
`=>D`

Similarity, SMB-013

Libby plays on a hockey field that is 120 metres long.

She makes a scale diagram of the field using a ratio of `1: 400`.
 

How long, in centimetres, should Libby make the scale diagram of the field?   (2 marks)

Show Answers Only

`text(30 cm)`

Show Worked Solution

`text(Ratio is)\ \ 1:400`

`text{Converting to centimetres:}`

`text{120 m = 120 × 100 = 12 000 cm}`

`:.\ text(Length of the scale diagram)`

`=12\ 000 ÷ 400`

`= 30\ text(cm)`

Similarity, SMB-012

Poppy uses a photocopier to enlarge this picture.
 

   

The enlarged picture is 3 times as high and 3 times as wide as the original.

By what factor is the area of the picture increased?   (2 marks)

Show Answers Only

`text(9 times the area of the original)`

Show Worked Solution

`text{Method 1}`

`text{Dimensions increased by a factor of 3}`

`:.\ text{Area increased by a factor of}\ 3^2 = 9`
 

`text{Method 2}`

`text(Area of original picture)\ = 3 xx 5 = 15\ text(cm)^2`

`text(Area of enlarged picture)\ = 9 xx 15 = 135\ text(cm)^2`

`:.\ text(Factor)\ = 135/15 = 9\ text(times)`

Similarity, SMB-011

Kransky has a photo which is 25 cm wide and 10 cm high

He wants to enlarge it to make a poster with a width of 50 cm.

What will be the height of the poster?   (2 marks)

Show Answers Only

`text(20 cm)`

Show Worked Solution
`text(Scale factor)` `= text(new width)/text(old width)`
  `= 50/25`
  `= 2`

 

`:.\ text(New height)` `= 2 xx 10`
  `= 20\ text(cm)`

Similarity, SMB-010 MC

The actual body length of a beetle Brad has caught is 24 mm.

A scale drawing of the beetle is shown below.

What scale is used in the drawing?

  1. \(1\ \text{cm represents } 5\ \text{mm}\)
  2. \(1\ \text{cm represents } 2\ \text{mm}\)
  3. \(2\ \text{cm represents } 1\ \text{mm}\)
  4. \(5\ \text{cm represents } 1\ \text{mm}\)
Show Answers Only

\(B\)

Show Worked Solution
\(12\ \text{cm}\) \(: 24\ \text{mm}\)
\(120\ \text{mm}\) \(: 24\ \text{mm}\)
\(5\) \(: 1\)
\(1\ \text{cm}\) \(: 2\ \text{mm  (possible options)}\)

 
\(\Rightarrow B\)

Similarity, SMB-009

Juanita wanted to print a larger copy of her photo of a tree.

She increased the scale by a factor of 3.

She named the first picture Photo A and the second picture Photo B.
 

 If Photo B is 3.2 cm high, how high is Photo A in centimetres, correct to 1 decimal place?    (2 marks)

Show Answers Only

`1.1\ text(cm)`

Show Worked Solution

`text{Since Photo B is increased by a factor of 3,}`

`text(Height of Photo B)/text(Height of Photo A)` `=3`
`:.\ text(Height of Photo A)` `=text(Height of Photo B)/3`
  `=3.2/3`
  `=1.0666…`
  `=1.1\ text{cm  (1 d.p.)}`

Similarity, SMB-008 MC

The picture below shows a flower.
 

 
The picture is 2 cm wide. The actual flower is 40 cm wide.

What scale is used in the picture?

  1. 1 cm represents 80 cm
  2. 4 cm represents 20 cm
  3. 1 cm represents 20 cm
  4. 1 cm represents 2 cm
Show Answers Only

`C`

Show Worked Solution

`2:40\ \ =>\ \ 1:20`

`text(1 cm represents 20 cm)`

`=>C`

Similarity, SMB-007

Two circles have the same centre, `O`, as shown in the diagram below.
 

The radius of the small circle is  `2/3`  the radius of the large circle.

Arc `CD` is 21 cm and the angle between the lines `AC` and `BD` is 45°.

What is the length of the arc `RS` in centimetres?   (3 marks)

Show Answers Only

`text(42 cm)`

Show Worked Solution

`text{Circles are similar}`

`text(Arc)\ ST` `= 2/3 xx text(arc)\ CD`
  `= 2/3 xx 21`
  `= 14\ text(cm)`

 

`angleSOR` `= 180-45`
  `= 135^@`
  `= 3 xx angleSOT`

 

`:. text(Arc)\ RS` `= 3 xx text(arc)\ ST`
  `= 3 xx 14`
  `= 42\ text(cm)`

Similarity, SMB-006

Select the pair of similar triangles.

The triangles are not drawn to scale.   (2 marks)

Triangle A Triangle B  
 
Triangle C Triangle D  
 
Show Answers Only

`text(Triangles A and D)`

Show Worked Solution

`text(Similar triangles are equiangular.)`

`text(Calculate triangle angles of each option.)`

`text(Triangle A: 55, 55, 70)`

`text(Triangle B: 50, 65, 65)`

`text(Triangle C: 50, 55, 65)`

`text(Triangle D: 55, 55, 70)`

`:.\ text{Triangles A and D are similar.}`

Similarity, SMB-005

Trapeziums `ABCD` and `AEFG` are similar.
 

What is the distance from `E` to `F`?   (2 marks)

Show Answers Only

`2\ text(m)`

Show Worked Solution

`text{Since trapeziums are similar:}`

`(EF)/(BC)` `= (AG)/(AD)`
`(EF)/8` `= 3/12`
`EF` `= (3 xx 8)/12`
  `= 2\ text(m)`

Similarity, SMB-004

A traffic light is 2.4 m tall. Its shadow from a nearby floodlight is 3 m long.
 

What is the height of the floodlight?   (3 marks)

Show Answers Only

`text(4.0 m)`

Show Worked Solution

`text{Triangles share common angle and both have right angles}`

`=>\ text{Triangles are similar (equiangular)}`

`text{Since corresponding sides are in similar ratios:}`

`h/5` `= 2.4/3`
`:. h` `= (5 xx 2.4)/3`
  `= 4\ text(metres)`

Similarity, SMB-003

On a map, the distance between two towns is measured at 54 millimetres.

  1. The actual distance between the two towns is 16.2 kilometres. What is the scale of the map?  (1 mark)

  2. If two other towns on the same map are 9.2 centimetres apart, what is the actual distance between the two towns in kilometres?  (1 mark)

Show Answers Only
  1. `1: 300\ 000`
  2. `27.6\ text(km)`
Show Worked Solution

i.   `text{Convert both distance to the same unit (cm).}`

`text(54 mm)` `= 5.4\ text(cm)`
`text(16.2 km)` `= 16\ 200\ text(m)`
  `= 1\ 620\ 000\ text(cm)`

 

`:.\ text(Scale) \ \ 5.4\ ` `: 1\ 620\ 000`
`1\ ` `: 300\ 000`

 

ii.    `text(Actual distance)` `= 9.2 xx 300\ 000`
    `= 2\ 760\ 000\ text(cm)`
    `= 27\ 600\ text(m)`
    `= 27.6\ text(km)`

Similarity, SMB-002

A map is drawn to scale, on 1-cm paper, showing the position of a supermarket and a cinema. A reservoir is also shown.
 


 

It takes 10 minutes to walk in a straight line from the cinema to the supermarket at a constant speed of 3 km/h. Show that the scale of the map is 1 cm = 100 m.  (3 marks)

Show Answers Only

`text(See Worked Solutions)`

Show Worked Solution

`text(3 km/h = 3000 metres per 60 minutes)`

♦ Mean mark part 49%.

`text(In 10 minutes:)`

`text(Actual distance) = 3000 xx 10/60 = 500\ text(metres)`

`text(Distance on map = 5 cm)`

`:.\ text(Scale   5 cm)` `: 500\ text(metres)`
`text(1 cm)` `: 100\ text(metres)`

Similarity, SMB-001

The scale on a given map is `1:80\ 000`.

If the actual distance between two points is 3.4 kilometres, how far apart on the map would be the two points be, in centimetres?  (2 marks)

Show Answers Only

`4.25\ text(cm)`

Show Worked Solution

`text(Distance on map)`

`= text(real distance)/text(scale)`

`= text(3.4 km)/(80\ 000)`

`= text(3400 m)/(80\ 000)`

`= 0.0425\ text(m)`

`= 4.25\ text(cm)`

Measurement, STD1 M5 2022 HSC 5 MC

Two similar figures are shown.
 

What is the value of `x` ?

  1. 6
  2. 8
  3. 18
  4. 27
Show Answers Only

`C`

Show Worked Solution

`text{Scale factor}\ =3/2 =1.5`

`:.\ x = 1.5 xx 12 = 18`
  

`text{Alternate solution}`

`text{Using sides of similar figures in the same ratio:}`

`x/12` `=3/2`  
`x` `=12 xx (3/2)`  
`x` `=18`  

   
`=> C`

Measurement, STD1 M5 2021 HSC 26

The diagrams show two similar shapes. The dimensions of the small shape are enlarged by a scale factor of 1.5 to produce the large shape.
 


 

Calculate the area of the large shape.  (3 marks)

Show Answers Only

`279\ text(cm)^2`

Show Worked Solution

`text(Dimension of larger shape:)`

♦♦ Mean mark 32%.

`text(Width) = 16 xx 1.5 = 24\ text(cm)`

`text(Height) = 9 xx 1.5 = 13.5\ text(cm)`

`text(Triangle height) = 2.5 xx 1.5 = 3.75\ text(cm)`

`:.\ text(Area)` `= 24 xx (13.5-3.75) + 1/2 xx 24 xx 3.75`
  `= 279\ text(cm)^2`

Measurement, STD1 M5 2020 HSC 28

Two similar right-angled triangles are shown.
 


 

The length of side `AB` is 8 cm and the length of side `EF` is 4 cm.

The area of triangle `ABC` is 20 cm2.

Calculate the length in centimetres of side `DF` in Triangle II, correct to two decimal places.   (4 marks)

Show Answers Only

`7.55\ \text{cm}`

Show Worked Solution

`text{Consider} \ Δ ABC :`

`text{Area}` `= frac{1}{2} xx AB xx BC`
`20` `= frac{1}{2} xx 8 xx BC`
`therefore \ BC` `= 5`

 

`text{Using Pythagoras in} \ Δ ABC :`

♦♦♦ Mean mark 11%.

`AC = sqrt(8^2 + 5^2) = sqrt89`

 

`text{S} text{ince} \ Δ ABC\ text{|||}\ Δ DEF,`

`frac{AC}{BC}` `= frac{DF}{EF}`
`frac{sqrt89}{5}` `= frac{DF}{4}`
`therefore \ DF` `= frac{4 sqrt89}{5}`
  `= 7.547 …`
  `= 7.55 \ text{cm (to 2 d.p.)}`

Plane Geometry, 2UA 2018 HSC 13b

In `Delta ABC`, sides `AB` and `AC` have length 3, and `BC` has length 2. The point `D` is chosen on `AB` so that `DC` has length 2.
 

  1. Prove that `Delta ABC` and `Delta CBD` are similar.  (2 marks)

  2. Find the length `AD`.  (2 marks)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `5/3`
Show Worked Solution

i.    `text(Prove)\ \ Delta ABC\ text(|||)\ Delta CBD`

`Delta ABC\ text{is isosceles:}`

`/_ ABC = /_ ACB qquad text{(angles opposite equal sides)}`

`Delta CBD\ text{is isosceles:}`

`/_ CBD = /_ CDB qquad text{(angles opposite equal sides)}`

 
`text{Since}\ \ /_ ABC =  /_ CBD`

`:. Delta ABC\ text(|||)\ Delta CDB qquad text{(equiangular)}`
 

ii.   `text(Using ratios of similar triangles)`

`(DB)/(CB)` `= (BC)/(AC)`
`{(3-AD)}/2` `= 2/3`
`3-AD` `= 4/3`
`:. AD` `= 5/3`

 

Measurement, STD2 M7 2016 HSC 16 MC

The width (`W`) of a river can be calculated using two similar triangles, as shown in the diagram.
  

What is the approximate width of the river?

  1. `17.8\ text(m)`
  2. `19.3\ text(m)`
  3. `23.2\ text(m)`
  4. `24.9\ text(m)`
Show Answers Only

`=> A`

Show Worked Solution

`text{Triangles are similar (equiangular)}`

`text(Using similar ratios:)`

`W/(7.1)` `= 20.3/8.1`
`:. W` `= (20.3 xx 7.1)/8.1`
  `= 17.79…`

 
`=> A`

Measurement, STD2 M7 2015 HSC 27a

At a particular time during the day, a tower of height 19.2 metres casts a shadow. At the same time, a person who is 1.65 metres tall casts a shadow 5 metres long.

  

What is the length of the shadow cast by the tower at that time?  (2 marks)

Show Answers Only

`58\ text{m}`

Show Worked Solution

`text(Both triangles have right angles and a common)`

`text(angle to the ground.)`

`:.\ text{Triangles are similar (equiangular)}`

 

`text(Let)\ x =\ text(length of tower shadow)`

`x/5` `= 19.2/1.65\ \ text{(corresponding sides of similar triangles)}`

 
`x` `= (5 xx 19.2)/1.65`  
  `= 58.1818…`  
  `= 58\ text{m  (nearest m)}`  

Measurement, STD2 M7 2007 HSC 4 MC

What scale factor has been used to transform Triangle `A` to Triangle `B`?
  

  1. `1/2`
  2. `3/4`
  3. `2`
  4. `3`
Show Answers Only

`A`

Show Worked Solution

`text(Take two corresponding sides)`

`text(In)\ Delta A:\ 3\ text(cm)`

`text(In)\ Delta B:\ 1 \frac{1}{2}\ text(cm)`

`:.\ text(Scale factor converting)\ Delta A\ text(to)\ Delta B = frac{1}{2}`

`=>  A`

Measurement, STD2 M7 2008 HSC 20 MC

A point `P` lies between a tree, 2 metres high, and a tower, 8 metres high. `P` is 3 metres away from the base of the tree.

From `P`, the angles of elevation to the top of the tree and to the top of the tower are equal.
 

What is the distance, `x`, from `P` to the top of the tower?

  1. 9 m
  2. 9.61 m
  3. 12.04 m
  4. 14.42 m
Show Answers Only

`D`

Show Worked Solution

`text(Triangles are similar)\ \ text{(equiangular)}`

`text(In smaller triangle:)`

`h^2` `= 2^2 + 3^2`
  `= 13`
`h` `= sqrt 13`
   
`x/sqrt13` `= 8/2\ \ \ text{(sides of similar Δs in same ratio)}`
`x` `= (8 sqrt 13)/2`
  `= 14.422…`

 
`=>  D`

Measurement, STD2 M7 2012 HSC 28c

Jacques and a flagpole both cast shadows on the ground. The difference between the lengths of their shadows is 3 metres.
 

What is the value of `d`, the length of Jacques’ shadow?     (3 marks)

Show Answers Only

 `d = 1.8\  text(m)`

Show Worked Solution
♦♦ Mean mark 24%

`text{Both triangles have right-angles with a common (ground) angle.}`

`:.\ text{Triangles are similar (equiangular)}`
 

` text{Since corresponding sides are in the same ratio}`

`d/1.5` `= (d+3)/4`
`4d` `= 1.5(d + 3)`
`8d` `= 3(d + 3)`
  `= 3d + 9`
`5d` `= 9`
`:.d` `= 9/5`
  `=1.8\ text(m)`

Measurement, STD2 M7 2012 HSC 27c

A map has a scale of  1 : 500 000.

  1. Two mountain peaks are 2 cm apart on the map.

     

    What is the actual distance between the two mountain peaks, in kilometres?  (1 mark)

  2. Two cities are 75 km apart. How far apart are the two cities on the map, in centimetres?   (1 mark)

Show Answers Only
  1. `text(10 km)`
  2. `text(15 cm)`
Show Worked Solution
♦ Mean mark 37%
MARKER’S COMMENT: Better responses realised that 1 unit on the map represented 1 unit x 500,000 in real life.
i.  `text{Actual distance (2 cm)}` `= 2 xx 500\ 000`
  `= 1\ 000\ 000\ text(cm)`
  `= 10\ 000\ text(m)`
  `=10\ text(km)`

 

`:.\ text(The 2 mountain peaks are 10 km apart.)`

 

♦ Mean mark 44%

ii.   `text(Cities are 75 km apart.)`

`text{From part (i), we know 2 cm = 10 km}`

`=>\ text(1 cm = 5 km)`

`=>\ text(On the map,  75 km)= 75/5=15\ text(cm)` 

`:.\ text(Distance on the map is 15 cm.)`

Measurement, STD2 M1 2013 HSC 17 MC

Triangles  `ABC`  and  `DEF`  are similar.
  

Which expression could be used to find the value of  `x`?

  1. `yxx10/15`
  2. `yxx10/23`
  3. `yxx15/10`
  4. `yxx23/15`
Show Answers Only

`C`

Show Worked Solution
♦ Mean mark 38%

`text(We know)\ \ Delta ABC\ text(|||)\ Delta DEF`

`:.\ (AB)/(AC)` `=y/10=(DE)/(DF)=x/15`
`x/15` `=y/10`
`x` `=yxx15/10`

 
`=> C`