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Pre-RMO 2014/3

Let \(ABCD\) be a convex quadrilateral with perpendicular diagonals. If \(AB = 20, BC = 70,\) and \(CD = 90\), then what is the value of \(DA\)?


This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
2 years, 11 months ago

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Let the diagonals meet at a point \(E\), and let \(EA=a, EB=b, EC=c, ED=d\)

Applying PT, we get

\(a^{2}+b^{2}=20^{2} \rightarrow Eq.1\)

\(b^{2}+c^{2}=70^{2} \rightarrow Eq.2\)

\(c^{2}+d^{2}=90^{2} \rightarrow Eq.3\)

Eq.3 - Eq.2

\(d^{2}-b^{2}=90^{2} -70^{2}\rightarrow Eq.4\)

Eq.1 + Eq.4

\(a^{2}+d^{2}=90^{2} -70^{2}+20^{2}\)

\( \Rightarrow a^{2}+d^{2}=3600=AD^{2}\)

\( \Rightarrow AD=60\) Aneesh Kundu · 2 years, 11 months ago

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Its \(\boxed{60}\) Akshat Sharda · 2 years, 1 month ago

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60 Rahul Verma · 2 years, 11 months ago

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60 Sahil Nare · 2 years, 4 months ago

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just a small correction a^2+b^2= 20^2 Aayush Patni · 2 years, 10 months ago

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@Aayush Patni Thnx edited. Aneesh Kundu · 2 years, 10 months ago

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60 Atharva Sarage · 2 years, 11 months ago

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60 Aditya Vimal · 2 years, 11 months ago

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60 Pooja Deshmukh · 2 years, 11 months ago

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60 Gaurav Singh · 2 years, 1 month ago

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Ans underroot 140 Aman Real · 2 years, 11 months ago

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@Aman Real Only an integer answer is possible. Sai Prasanth Rao · 2 years, 11 months ago

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