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Similarity, SMB-016

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

Find the area of Triangle II?  (3 marks)

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`24\ text{cm}^2`

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`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-006

Select the pair of similar triangles.

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

Triangle A Triangle B  
 
Triangle C Triangle D  
 
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`text(Triangles A and D)`

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`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-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, 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)

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`7.55\ \text{cm}`

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`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)

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  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `5/3`
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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 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`
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`C`

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♦ 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`