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In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that:

(i) ΔAOB is similar to ΔCOD.

(ii) OA x OD = OB x OC.

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Answer

Quadrilateral ABCD is shown in the figure below:

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that (i) ΔAOB is similar to ΔCOD. (ii) OA x OD = OB x OC. Similarity, Concise Mathematics Solutions ICSE Class 10.

(i) Given,

AO = 2CO and BO = 2DO

⇒ AOCO=21 and BODO=21\dfrac{AO}{CO} = \dfrac{2}{1} \text{ and } \dfrac{BO}{DO} = \dfrac{2}{1}

∴AOCO=BODO\therefore \dfrac{AO}{CO} = \dfrac{BO}{DO}

From figure,

∠AOB = ∠DOC [Vertically opposite angles are equal].

∴ ∆AOB ~ ∆COD [By S.A.S.]

Hence, proved that ∆AOB ~ ∆COD.

(ii) Since,

AOCO=BODO\dfrac{AO}{CO} = \dfrac{BO}{DO} [Proved above]

∴ OA x OD = OB x OC.

Hence, proved that OA x OD = OB x OC.

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