Triangle ABC and DEF are similar. What is the length of DF?*

We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

∴
area(△DEF)
area(△ABC)

=(
EF
BC

)
2

⇒
25
16

=(
EF
2.3

)
2

⇒
5
4

=
EF
2.3


⇒ EF=
4
2.3×5


∴ EF=2.875cm

(ii) We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

∴
area(△DEF)
area(△ABC)

=(
DE
AB

)
2

⇒
64
9

=(
DE
AB

)
2

⇒
8
3

=
5.1
AB


∴ AB=1.91cm

(iii) We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

∴
area(△DEF)
area(△ABC)

=(
DF
AC

)
2

∴
area(△DEF)
area(△ABC)

=(
8
19

)
2

∴
area(△DEF)
area(△ABC)

=(
64
361

)

(iv) We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides

∴
area(△DEF)
area(△ABC)

=(
DE
AB

)
2

⇒
64
36

=(
DE
AB

)
2

⇒
8
6

=
6.2
AB


∴ AB=4.65cm

(v) We know that the ratio of area of two similar triangles is equal to the ratio of squares of their corresponding sides.

∴
area(△DEF)
area(△ABC)

=(
DE
AB

)
2

∴
area(△DEF)
area(△ABC)

=(
1.4
1.2

)
2

∴
area(△DEF)
area(△ABC)

=
49
36