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Question

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?

A
2:7
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B
3:5
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C
4:7
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D
2:5
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E
3:7
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Solution

The correct option is D 2:5

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:

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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