Q. 55.0( 1 Vote )

□ABCD is a rhombus. = {0}. Prove that the area of ΔOAB = (area of □ ABCD).

Given: ABCD is a rhombus. = {0}.

To prove: Area of OAB = (area of ABCD)

Proof:

O is the mid-point of AC as well as BD.

Further, in rhombus ABCD,

AB BC CD DA

In ∆OAB and ∆OBC,

OA OC

OB OB

AB CB

∆OAB and ∆OBC are congruent by SSS theorem for congruence.

Thus, their areas are equal.

Similarly, ∆OAB, ∆OBC, ∆OCD and ∆ODA are all congruent triangles having equal areas.

∆OAB = ∆OBC = ∆OCD = ∆ODA

Now, ABCD = ∆OAB + ∆OBC + ∆OCD + ∆ODA

ABCD = ∆OAB + ∆OAB + ∆OAB + ∆OAB

ABCD = 4∆OAB

Thus,

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