Question

In ∆ABC, ∠B = 90° and BD ⊥ AC.

1. If CD = 10 cm and BD = 8 cm; find AD.
2. If AC = 18 cm and AD = 6 cm; find BD.
3. If AC = 9 cm and AB = 7 cm; find AD.

Answer 1

i. In ∆CDB,

∠1 + ∠2 + ∠3 = 180°

∠1 + ∠3 = 90°   ...(1) (Since, ∠2 = 90°)

∠3 + ∠4 = 90°   ...(2) (Since, ∠ABC = 90°)

From (1) and (2),

∠1 + ∠3 = ∠3 + ∠4

∠1 = ∠4

Also, ∠2 = ∠5 = 90°

∴ ∆CDB ~ ∆BDA   ...(By AA similarity)

`=> (CD)/(BD) = (BD)/(AD)`

`=>` BD² = AD × CD

`=>` (8)² = AD × 10

`=>` AD = 6.4

Hence, AD = 6.4 cm

ii. Also, by similarity, we have:

`(BD)/(DA) = (CD)/(BD)`

BD² = 6 × (18 – 6)

BD² = 72

Hence, BD = 8.5 cm

iii. Clearly, ∆ADB ~ ∆ABC

`∴(AD)/(AB)=(AB)/(AC)`

`AD = (7 xx 7)/9`

= `(49)/9`

= `5 4/9`

Hence, `AD = 5 4/9 cm`