Question

(b) In the given figure, DE || BC.
(i) Prove that ∆ADE and
∆ABC are similar.
DE
1
(ii) Given that AD
BD,
2
calculate DE, if BC = 4.5 cm.
B В
С
(iii) Find the ratio of the areas
of ∆ADE and ∆ABC.​

Answer 1

Solution :-

from image, we have given that, DE || BC .

so, in ∆ADE and ∆ABC we have,

→ ∠ADE = ∠ABC { since DE || BC and AB is a transversal .}

→ ∠DAE = ∠BAC { common .}

then ,

→ ∆ADE ~ ∆ABC { By AA similarity. }

now, we know that, DE || BC, by BPT ,

• AD / AB = AE / AC = DE / BC .

given that,

• AD = (1/2)BD .

so,

→ AD = (1/2)BD

→ AD / BD = (1/2)

then,

→ AD / AB = AD / (AD + BD)

→ AD / AB = 1 / (1 + 2)

→ AD / AB = 1/3 .

also given,

• BC = 4.5 cm.

therefore,

→ DE / BC = AD / AB

→ DE / BC = 1/3

→ DE / 4.5 = 1/3

→ DE = 4.5/3 = 1.5 cm.

now, we know that, when two ∆'s are similar,

• Ratio of areas of ∆'s = Ratio of square of their corresponding sides .

hence,

→ Area ∆ADE : Area ∆ABC = AD² : AB² = 1² : 3² = 1 : 9

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