Question
Two isosceles triangles have equal vertical angles. Show that the triangles are similar.If the ratio between the areas of these two triangles is $16 : 25,$ find the ratio between their corresponding altitudes.
✓
Answer
Let ABC and PQR be two isosceles triangles.
Then $\frac{A B}{A C}=\frac{1}{1}$ And $\frac{P Q}{P R}=\frac{1}{1}$
Also, $\angle A = \angle P$ (Given)
$\therefore \triangle ABC ~ \triangle PQR$ (SAS similarity)
Let AD and PS be the altitude in the respective triangles.
We know that the ratio of areas of two similar triangles is equal to the square of their corresponding altitudes.
$\frac{A r(\triangle A B C)}{A r(\triangle P Q R)}=\left(\frac{A D}{P S}\right)^2$
$\frac{16}{25}=\left(\frac{A D}{P S}\right)^2$
$\frac{A D}{P S}=\frac{4}{5}$
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