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Question

In a right triangle ΔABC with right angle at B and BD is a perpendicular dropped onto the hypotenuse. If AC = 2AB, what is the area of ΔABD? (area of ΔABC = 5 sq units.)

A

0.5 sq. units

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B

1 sq. unit

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C

1.25 sq. units

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D

5 sq. units

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Solution

The correct option is C 1.25 sq. units Given,ABAC=12 and Ar(Δ ABC) = 5 sq.units Consider ΔABD and ΔACB: ∠BAD=∠CAB ----[common angle] ∠BDA=∠CBA ----[90∘] By AA similarity criterion,ΔABD∼ΔABC Hence, Ar(ΔABD)Ar(ΔACB)=(ABAC)2=(12)2=14 i.e., Ar(ΔABD)×4=Ar(ΔACB)⇒Ar(ΔABD)=54=1.25 sq.units Hence, the area of ΔABD is 1.25 sq.units.

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