Question

19. In A ABC, ZB = 90° and BD I AC.
(i) If CD = 10 cm and BD = 8 cm; find AD.
(ii) If AC = 18 cm and AD = 6 cm; find BD.
(iii) If AC = 9 cm and AB = 7 cm; find AD.
With​

Answer 1

Answer:

We have,     ∠ABC = 90°

⇒∠ABD + ∠DBC = 90°      .....(1)

In ΔBDC,

∠BDC + ∠DCB + ∠DBC = 180°   (Angle sum property)

⇒90° + ∠DCB + ∠DBC = 180°  

⇒∠DCB + ∠DBC = 90°    .....(2)

Now, from (1) and (2), we get

∠ABD + ∠DBC = ∠DCB + ∠DBC

⇒∠ABD = ∠DCB    ......(3)

In ΔADB and ΔBDC,

∠ABD = ∠DCB   [Using (3)]

∠ADB = ∠BDC  [90° each]

ΔADB ~ ΔBDC [AA similarity]

⇒   [Corresponding sides of similar Δ's are proportional]

⇒

⇒   ........(4)

Now, put CD = 10 cm; BD = 8 cm in (4), we get    

⇒ AD = 6.4 cm

In ΔADB and ΔABC,      ∠A = ∠A   [Common]

∠ADB = ∠ABC   [90° each]

ΔADB ~ ΔABC   (AA similarity)

⇒   (Corresponding sides of similar Δ's are proportional)

⇒

⇒   .....(5)

Put AD = 6 cm; AC = 18 cm in (5), we get

⇒

⇒

In ΔADB,        [Pythagoras theorem]

⇒

⇒

⇒BD = = 8.5 cm

Now, put AC = 9 cm; AB = 7 cm in (5), we get    

⇒AD =

⇒AD = = 5.4 cm