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Q.

Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Then the ratio of their corresponding heights is,


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a

4:3

b

4:7

c

2:5

d

4:5 

answer is D.

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Detailed Solution

Given that,
Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25.
In  the triangle Δ  ABC and ΔDEF.
Consider the figure,
Question ImageIt is given that ∠  A =∠  D, DE=DF  , and AB=AC.  
Therefore,
ΔABC  and ΔDEF are similar by SAS pattern.
 SO, the corresponding angles are in the same ratio.
So, ∠  B =∠  E and∠  C =∠  F .
In the triangles Δ  ABM and ΔDEN, It is known that ∠  B =∠  E. As the angle M and angle N are right angle.
So, both angles are equal that is ∠  M =∠  N.
So, the triangles Δ  ABM andΔ  DEN are similar by the AA pattern.
Then,
AM DN = AB DE = BM EN .  
As the triangles Δ  ABC andΔ  DEF are similar then the ratio of two similar triangles is equal to the square of the ratio of their corresponding sides,
  ar ΔABC ar ΔDEF = AB 2 DE 2 ⇒ 16 25 = AM 2 DN 2 ⇒ AM DN = 4 2 5 2 ⇒ AM DN = 4 5  
The ratio of their corresponding heights of AM DN = 4 5 .  
Hence, option 4 is correct.
 
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