Question

In the given figure, DE || BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm.

1. Write all possible pairs of similar triangles.
2. Find lengths of ME and DM.

Answer 1

i. In ΔAME and ΔANC,

∠AME = ∠ANC   ...(Since DE || BC that is, ME || NC)

∠MAE = ∠NAC  ...(Common angle)

`=>` ΔAME and ΔANC  ...(AA criterion for similarity)

In ΔADM and ΔABN,

∠ADM = ∠ABN   ...(Since DE || BC that is, ME || BN)

∠DAM = ∠BAN  ...(Common angle)

`=>` ΔADM and ΔABN   ...(AA criterion for similarity)

In ΔADE and ΔABC,

∠ADE = ∠ABC   ...(Since DE || BC that is, ME || NC)

∠AED = ∠ACB  ...(Since DE || BC)

`=>` ΔADE and ΔABC  ...(AA criterion for similarity)

ii. In ΔAME and ΔANC, 

∠AME = ∠ANC ...(Since DE || BC that is, ME || NC)

∠MAE = ∠NAC ...(Common angle)

`=>` ΔAME and ΔANC   ...(AA criterion for similarity)

`=> (ME)/(NC) = (AE)/(AC)`

`=> (ME)/6 = 15/24`

`=>` ME = 3.75 cm

In ΔADE and ΔABC, 

∠ADE = ∠ABC  ...(Since DE || BC that is, ME || NC)

∠AED = ∠ACB  ...(Since DE || BC)

`=>` ΔADE and ΔABC   ...(AA criterion for similarity)

`=> (AD)/(AB) = (AE)/(AC) = 15/24`  ...(i)

In ΔADM and ΔABN, 

∠ADM = ∠ABN   ...(Since DE || BC that is, ME || NC)

∠DAM = ∠BAN  ...(Common angle)

`=>` ΔADM and ΔABN   ...(AA criterion for similarity)

`=> (DM)/(BN) = (AD)/(AB) = 15/24`  ...(From (i))

`=> (DM)/24 = 15/24`

`=>` DM = 15 cm