Two isosceles Δs have equal vertical angles and their areas are in the ratio 9 : 16. Find the ratio of their corresponding heights (altitudes).
                         

Given: ΔABC and ΔPQR are isosceles and ∠A = ∠P. AD, PS are the altitudes and .
To find: AD
          PS
Proof: In Δ ABC, ∠B = ∠C (isosceles Δ property) Similarly in Δ PQR, ∠Q = ∠R.
                  ∠A = ∠P (given)∴ ∠B = ∠C =
Since ∠ A = ∠ P 
∠ B = ∠ C = ∠ Q = ∠R 
∴ Δ ABC ~ Δ PQR (AA)
If 2 triangles are similar then the ratio of areas will be equal to the square of the corresponding sides,  
…………………(1)
In Δ ABD, Δ PQS
∠D = ∠S (= 90° )
∠B = ∠Q (given)
∴ Δ ABD ~ Δ PQS (AA)
∴  AB = AD  ……………….(2) 
  PQ    PS
According equation (1)
∴  
⇒ AD  =  3
   PS     4
.^.. The ratio of their corresponding heights is 3 : 4.