Given :- In the fig, AB || CD, FIND x.(Hint:Prove that AOB ~ COD).

Solution :-

in ∆AOB and ∆COD , we have,

→ AB || CD .

So,

→ ∠ABO = ∠CDO { ALTERNATE angles. }

similarly,

→ ∠BAO = ∠DCO { Alternate angles. }

and,

→ ∠AOB = ∠COD { vertically opposite angles. }

Therefore,

→ ∆AOB ~ ∆COD { By AAA similarity. }

Hence,

→ AO / CO = BO / DO { in similar ∆'s , ratio of corresponding sides are equal. }

PUTTING values now, we get,

→ 4 / (4x - 2) = (x + 1) / (2x + 4)

cross - Multiply,

→ 4(2x + 4) = (4x - 2)(x + 1)

→ 8x + 16 = 4x² + 4x - 2x - 2

→ 4x² + 2x - 8x - 2 - 16 = 0

→ 4x² - 6x - 18 = 0

→ 2(2x² - 3x - 9) = 0

→ 2x² - 3x - 9 = 0

→ 2x² - 6x + 3x - 9 = 0

→ 2x(x - 3) + 3(x - 3) = 0

→ (x - 3)(2x + 3) = 0

Putting both equal to 0, we get,

→ x = 3 or , (-3/2)

since negative VALUE of x is not Possible.

→ x = 3 (Ans.)

Hence, x is equal to 3.

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