smc-1105-30-Similarity

v1 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)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`94.5\ text(cm)^2`

Show Worked Solution

`text(Dimension of larger shape:)`

♦♦ Mean mark 32%.

`text(Width) = 6 xx 1.5 = 9\ text(cm)`

`text(Height) = 8 xx 1.5 = 12 \ text(cm)`

`text(Triangle height) = 2 xx 1.5 = 3\ text(cm)`

`:.\ text(Area)`	`= 9 xx (12-3) + 1/2 xx 9 xx 3`
	`= 94.5\ text(cm)^2`

v1 Measurement, STD2 M7 2008 HSC 20 MC

A point `P` lies between a lamp post, 1.5 metres high, and a building, 7 metres high. `P` is 2.5 metres away from the base of the post.

From `P`, the angles of elevation to the top of the lamp post and to the top of the building are equal.

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

10.60
12.50
13.60
14.55

Show Answers Only

`C`

Show Worked Solution

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

`text(In smaller triangle:)`

`h^2`	`= 1.5^2 + 2.5^2`
	`= 8.5`
`h`	`= sqrt 8.5`

`x/sqrt8.5`	`= 7/1.5 \ \ \ text{(sides of similar Δs in same ratio)}`
`x`	`= (7 sqrt 8.5)/1.5`
	`= 13.60…`

`=> C`

v1 Measurement, STD2 M7 2016 HSC 16 MC

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

What is the approximate width of the road?

`12.8\ text(m)`
`13.3\ text(m)`
`14.6\ text(m)`
`17.8\ text(m)`

Show Answers Only

`=> C`

Show Worked Solution

`text{Triangles are similar (equiangular)}`

`text(Using similar ratios:)`

`W/(6.5)`	`= 18/8`
`:. W`	`= (18 xx 6.5)/8`
	`= 14.62…`

`=> C`

Measurement, STD1 M5 2022 HSC 5 MC

Two similar figures are shown.

What is the value of `x` ?

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)

--- 5 WORK AREA LINES (style=lined) ---

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)

--- 8 WORK AREA LINES (style=lined) ---

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.)}`

Measurement, STD1 M5 2019 HSC 10 MC

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

What is the area of Triangle II?

18 cm²
24 cm²
30 cm²
48 cm²

Show Answers Only

`B`

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`

`=> B`

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?

`17.8\ text(m)`
`19.3\ text(m)`
`23.2\ text(m)`
`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)

--- 4 WORK AREA LINES (style=lined) ---

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`?

`1/2`
`3/4`
`2`
`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?

9 m
9.61 m
12.04 m
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)

--- 6 WORK AREA LINES (style=lined) ---

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 M1 2013 HSC 17 MC

Triangles `ABC` and `DEF` are similar.

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

`yxx10/15`
`yxx10/23`
`yxx15/10`
`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`