Question

Two isosceles triangles have equal vertical angles and their areas are in the ratio 36.25 Find the ratio of their corresponding heights.

Answer 1

Let in $△$ ABC with height as AD,
AB = AC
⇒ $\angle$B = $\angle$C = 180- $\angle$ $\frac{A}{2}$ .... Equation-1
And in $△$PQR with height as PS
PQ = PR
⇒ $\angle$Q = $\angle$R = 180-∠ $\frac{P}{2}$ ...... Equation- 2
Given vertical angles of the two triangles are equal.
i.e., $\angle$A = $\angle$P
and $\angle$B = $\angle$C = $\angle$Q = $\angle$R
By AAA similarity criterion,
We know ratio between the areas of two similar triangle is same as the ratio between the square of their corresponding altitudes.
And corresponding heights of two given triangles are AD and PS.
$\frac{Areaof\Delta ABC}{Areaof\Delta PQR}=\frac{AD²}{PS²}$
=$\frac{36²}{25²}=\frac{AD²}{PS²}$
AD : PS = 6 : 5 ( ratio of their corresponding heights).