45 21. In AABC, DE || BC in given figure, then BD is equal to A 4 cm 6 cm D E 3 cm B B С C (a) 2 cm (6) 8 cm (c) 3 cm (d) 9 cm
Answers
Step-by-step explanation:
In △ABC, DE∥BC
⇒ ∠B=∠D [ Corresponding angles ]
⇒ ∠C=∠E [ Corresponding angles ]
⇒ ∠A=∠A [ Common angle]
∴ 4△ABC∼△ADE [ By AAA criteria ]
(i)
area(△ADE)
area(△ABC)
=(
DE
BC
)
2
[ Ratio of areas of two similar triangles is equal the ratio of squares of their corresponding sides. ]
∴
16
area(△ABC)
=(
4
6
)
2
∴ area(△ABC)=
16
36×16
∴ area(△ABC)=36cm
2
(ii)
area(△ADE)
area(△ABC)
=(
DE
BC
)
2
[ Ratio of areas of two similar triangles is equal the ratio of squares of their corresponding sides. ]
∴
25
area(△ABC)
=(
4
8
)
2
∴ area(△ABC)=
16
64×25
∴ area(△ABC)=100cm
2
(iii)
area(△ABC)
area(△ADE)
=(
BC
DE
)
2
[Ratio of areas of two similar triangles is equal the ratio of squares of their corresponding sides. ]
area(△ABC)
area(△ADE)
=(
5
3
)
2
area(△ABC)
area(△ADE)
=(
25
9
)
⇒
9
25
×area(△ADE)=area(△ABC)
⇒ AreaoftrapeziumBCED=area(△ABC)−area(△ADE)
∴
AreaofBCED
area(△ADE)
=
area(△ABC)−area(△ADE)
area(△ADE)
AreaofBCED =
(
9
25
−1)area(△ADE)
area(△ADE)
=
9
16
1
=
16
9