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In the given isosceles right angled triangle UVW, a square PQRS is inscribed as shown in the figure. If PV:VS=2:1, what is the ratio of areas of the square to the outer triangle UVW? In The Given Isosceles Right Angled Triangle UVW, A Square P...


Options:
A .   2:7
B .   3:5
C .   4:7
D .   2:5
E .   3:7
Answer: Option D
:
D

Ans: d. 2:5


Using variables


In the given figure, we draw QT || VW.


Δ PTQ and Δ PVS are congruent (A,A,A and side)


Hence PT=Y, QT=X=UT (also since ΔUTQ and ΔUVW are similar).


Thus UV=UW=2X+Y. Area of square: X2+Y2; Area of  ΔUVW= 12 *(2X+Y)2.


Given X=2Y, X2+Y2: ( 12 *(2X+Y)2) = 5Y2: ( 25Y2* 12 ); = 25.


 Using numbers,You can solve the problem faster as follows:


In The Given Isosceles Right Angled Triangle UVW, A Square P...


PV=2, SV=1 => PS= 5 Area of square =5


PTQ congruent to PVS


PT=1 and QT=2


UT=TQ (45-45-90) UT= 2


Now UV= 5 WV=5 (45-45-90)


UW=52


 


(Area of square / Area of triangle  )
= (5(25/2)) = 2:5

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