Question

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

Answer 1

In quadrilateral ABCD, diagonals AC and BD intersect each other at O such that

AO = 2CO and BO = 2DO

In triangles △AOB and △COD,

`(AO)/(CO) = 2 and (BO)/(DO) = 2`

Hence,

`(AO)/(CO) = (BO)/(DO)`

Also, diagonals AC and BD intersect at O, so ∠AOB = ∠COD (as vertically opposite angles).

Therefore, in triangles △AOB and △COD,

The two sides are proportional, and the included angle is equal (by the SAS similarity criterion).

∴ △AOB ∼ △COD

Hence proved.