I want to know the best answer of the question from Areas of Parallelograms and Triangles chapter of class 9^{th} ncert math. The question from exercise 9.3of math. Give me the easy way for solving this question of 6(1) In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that: (i) ar (DOC) = ar (AOB)

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# In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that: (i) ar (DOC) = ar (AOB) Q.6(1)

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Given,

OB = OD and AB = CD

Construction,

DE âŠ¥ AC and BF âŠ¥ AC are drawn.

Proof:

(i) In Î”DOE and Î”BOF,

âˆ DEO = âˆ BFO (Perpendiculars)

âˆ DOE = âˆ BOF (Vertically opposite angles)

OD = OB (Given)

âˆ´, Î”DOE â‰… Î”BOF by AAS congruence condition.

âˆ´, DE = BF (By CPCT) â€” (i)

also, ar(Î”DOE) = ar(Î”BOF) (Congruent triangles) â€” (ii)

Now,

In Î”DEC and Î”BFA,

âˆ DEC = âˆ BFA (Perpendiculars)

CD = AB (Given)

DE = BF (From i)

âˆ´, Î”DEC â‰… Î”BFA by RHS congruence condition.

âˆ´, ar(Î”DEC) = ar(Î”BFA) (Congruent triangles) â€” (iii)

Adding (ii) and (iii),

ar(Î”DOE) + ar(Î”DEC) = ar(Î”BOF) + ar(Î”BFA)

â‡’Â ar (DOC) = ar (AOB)