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

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

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TopNewestLet 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, 7 months ago

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60 – Rahul Verma · 2 years, 7 months ago

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Its \(\boxed{60}\) – Akshat Sharda · 1 year, 9 months ago

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60 – Sahil Nare · 2 years, 1 month ago

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just a small correction a^2+b^2= 20^2 – Aayush Patni · 2 years, 6 months ago

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– Aneesh Kundu · 2 years, 6 months ago

Thnx edited.Log in to reply

60 – Atharva Sarage · 2 years, 7 months ago

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60 – Aditya Vimal · 2 years, 7 months ago

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60 – Pooja Deshmukh · 2 years, 7 months ago

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60 – Gaurav Singh · 1 year, 9 months ago

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Ans underroot 140 – Aman Real · 2 years, 7 months ago

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– Sai Prasanth Rao · 2 years, 7 months ago

Only an integer answer is possible.Log in to reply