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Question Number 86907 by TawaTawa1 last updated on 01/Apr/20

Commented by mr W last updated on 01/Apr/20

question is wrong !? there are infinite possibilities for PQRS. (([PQRS])/(ΔABC))≠constant, but min.=(2/5)

questioniswrong!?thereareinfinitepossibilitiesforPQRS.[PQRS]ΔABC≠constant,butmin.=25

Commented by mr W last updated on 01/Apr/20

or you didn′t show the complete question.

oryoudidn′tshowthecompletequestion.

Answered by mr W last updated on 01/Apr/20

let AB=BC=a Δ_(ABC) =(a^2 /2) let BQ=x, BP=y, PQ=s s^2 =x^2 +y^2 AP=a−y ((PS)/(sin 45°))=((AP)/(sin (45°+∠APS)))=((AP)/(sin (45°+180°−90°−∠BPQ))) (s/(sin 45°))=((a−y)/(sin (45°+90°−∠BPQ))) (s/(sin 45°))=((a−y)/(cos (45°−∠BPQ))) s=((a−y)/(cos ∠BPQ+sin ∠BPQ)) ((a−y)/s)=cos ∠BPQ+sin ∠BPQ=(x/s)+(y/s) ⇒x+2y=a there are infinite possibilities: (a/3)≤y≤(a/2) [PQRS]=s^2 =x^2 +y^2 =(a−2y)^2 +y^2 (([PQRS])/(ΔABC))=((2s^2 )/a^2 )=((2[(a−2y)^2 +y^2 ])/a^2 )=2[(1−2(y/a))^2 +((y/a))^2 ] min(([PQRS])/(ΔABC))=(2/5) when y=((2a)/5), x=(a/5).

letAB=BC=aΔABC=a22letBQ=x,BP=y,PQ=ss2=x2+y2AP=a−yPSsin45°=APsin(45°+∠APS)=APsin(45°+180°−90°−∠BPQ)ssin45°=a−ysin(45°+90°−∠BPQ)ssin45°=a−ycos(45°−∠BPQ)s=a−ycos∠BPQ+sin∠BPQa−ys=cos∠BPQ+sin∠BPQ=xs+ys⇒x+2y=athereareinfinitepossibilities:a3⩽y⩽a2[PQRS]=s2=x2+y2=(a−2y)2+y2[PQRS]ΔABC=2s2a2=2[(a−2y)2+y2]a2=2[(1−2ya)2+(ya)2]min[PQRS]ΔABC=25wheny=2a5,x=a5.

Commented by TawaTawa1 last updated on 01/Apr/20

God bless you sir, i appreciate.

Godblessyousir,iappreciate.

Commented by mr W last updated on 01/Apr/20

you accepted this answer? but it doesn′t match the question.

youacceptedthisanswer?butitdoesn′tmatchthequestion.

Commented by TawaTawa1 last updated on 01/Apr/20

I agree sir. Thanks again.

Iagreesir.Thanksagain.

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