Question

In the given figure, ∠AMN = ∠MBC = 76°. If p, q and r are the lengths of AM, MB and BC respectively then express the length of MN in terms of p, q and r.  

 

Answer 1

Sol:
In ΔAMN and ΔABC
∠𝐴𝑀𝑁 = ∠𝐴𝐵𝐶 = 76°  (𝐺𝑖𝑣𝑒𝑛)
∠𝐴 = ∠𝐴 (𝐶𝑜𝑚𝑚𝑜𝑛)
By AA similarity criterion, ΔAMN ~ ΔABC
If two triangles are similar, then the ratio of their corresponding sides are proportional 

∴` (AM)/(AB)=(MN)/(BC)` 

⇒ `(AM)/(AM+MB)=(MN)/(BC)` 

⇒ `a/(a+b)=(MN)/c` 

⇒ `MN=(ac)/(a+b)`