Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. The ratio of their corresponding heights is ____.
Let ΔABC and ΔDEF be the given triangles in which
AB=AC,DE=DF and ∠A=∠D
Draw AL⊥BC and DM⊥EF
Now, ABAC=1 and DEDF=1
(∵ AB = AC and DE = DF)
⇒ABAC=DEDF
⇒ABDE=ACDF.....(1)
∴ In ΔABC and ΔDEF,
ABDE=ACDF [ From (1)]
and ∠A=∠D
⇒ΔABC∼ΔDEF (by SAS similarity criterion)
But, the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding heights.
Area (ΔABC)Area (ΔDEF)=AL2DM2
⇒1625=(ALDM)2
⇒ALDM=45
∴ AL : DM = 4 : 5 , i.e, the ratio of their corresponding heights is 4 : 5 .