Pre-RMO 2014/3

Let ABCDABCD be a convex quadrilateral with perpendicular diagonals. If AB=20,BC=70,AB = 20, BC = 70, and CD=90CD = 90, then what is the value of DADA?


This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
5 years ago

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Let the diagonals meet at a point EE, and let EA=a,EB=b,EC=c,ED=dEA=a, EB=b, EC=c, ED=d

Applying PT, we get

a2+b2=202Eq.1a^{2}+b^{2}=20^{2} \rightarrow Eq.1

b2+c2=702Eq.2b^{2}+c^{2}=70^{2} \rightarrow Eq.2

c2+d2=902Eq.3c^{2}+d^{2}=90^{2} \rightarrow Eq.3

Eq.3 - Eq.2

d2b2=902702Eq.4d^{2}-b^{2}=90^{2} -70^{2}\rightarrow Eq.4

Eq.1 + Eq.4

a2+d2=902702+202a^{2}+d^{2}=90^{2} -70^{2}+20^{2}

a2+d2=3600=AD2 \Rightarrow a^{2}+d^{2}=3600=AD^{2}

AD=60 \Rightarrow AD=60

Aneesh Kundu - 5 years ago

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60

Rahul Verma - 5 years ago

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Its 60\boxed{60}

Akshat Sharda - 4 years, 2 months ago

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60

Pooja Deshmukh - 5 years ago

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60

aditya vimal - 5 years ago

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60

Sahil Nare - 4 years, 5 months ago

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60

Atharva Sarage - 4 years, 12 months ago

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just a small correction a^2+b^2= 20^2

Aayush Patni - 4 years, 10 months ago

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Thnx edited.

Aneesh Kundu - 4 years, 10 months ago

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60

gaurav singh - 4 years, 2 months ago

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Ans underroot 140

Aman Real - 5 years ago

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Only an integer answer is possible.

Sai Prasanth Rao - 5 years ago

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