question_answer

Two isosceles triangles have equal angles and their areas are in the ratio\[\text{16}:\text{25}\]. The ratio of their corresponding altitudes is:

A) 4:5

B) 5:4

C) 3:2

D) 5:7

Correct Answer: A

Solution :

[a]

Let \[\Delta ABC\]and \[\Delta PQR\] be two isosceles triangles such that \[\angle A=\angle P,\] \[\angle B=\angle Q\] and \[\angle C=\angle R\]

\[\therefore \,\,\,\Delta ABC\tilde{\ }\Delta PQR\]

(By AAA similarity criterion)

Let AD and PS be the altitudes of \[\Delta ABC\] and \[\Delta PQR\]respectively,

Since, ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding altitudes.

\[\therefore \,\,\,\,\,\,\,\frac{ar(\Delta ABC)}{ar(\Delta PQR)}={{\left( \frac{AD}{PS} \right)}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\frac{16}{25}={{\left( \frac{AD}{PS} \right)}^{2}}\,\,\,\Rightarrow \,\,\,\frac{AD}{PS}=\frac{4}{5}\]