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Question

ABC is a right angled triangle, right angled at B. BD is a perpendicular to AC as shown in the figure. If AD:DC = 1:2, then AB2BC2 is ____.


A

1:2

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B

1:4

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C

4:1

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D

2:1

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Solution

The correct option is A

1:2


Consider ADB and ABC

∠BAD = ∠BAC [common angle]

∠BDA = ∠ABC [ 90]

Therefore by AA similarity criterion,
ADB and ABC are similar.

So, ABAC = ADAB

AB2 = AC. AD -------------(I)

Similarly, BDC and ABC are similar.

So, BCAC = DCBC

BC2 = AC. DC -------------(II)

Dividing (I) and (II) and cancelling out AC, we get
(ABBC)2 = ADDC -------------(III)

So, the ratio is the same, i.e., 1:2.


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