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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)

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`text(4.0 m)`

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`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)`

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)`
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`=> A`

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`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)

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`58\ text{m}`

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`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 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
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`D`

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`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)

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 `d = 1.8\  text(m)`

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♦♦ 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)`