Let \(XOY\) be a triangle with \(\angle XOY = 90^\circ\). Let \(M\) and \(N\) be the midpoints of legs \(OX\) and \(OY\), respectively. Suppose that \(XN = 19\) and \(YM = 22\). What is \(XY\)?

This note is part of the set Pre-RMO 2014

Let \(XOY\) be a triangle with \(\angle XOY = 90^\circ\). Let \(M\) and \(N\) be the midpoints of legs \(OX\) and \(OY\), respectively. Suppose that \(XN = 19\) and \(YM = 22\). What is \(XY\)?

This note is part of the set Pre-RMO 2014

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TopNewestis XY 26???? – Nitish Deshpande · 2 years ago

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Let \(XM=MO=a\) and \(ON=NY=b\)

By applying PT, we get

\(4a^{2}+b^{2}=19^{2} \rightarrow Eq.1\)

\(a^{2}+4b^{2}=22^{2} \rightarrow Eq.2\)

Eq.1+Eq.2

\(a^{2}+b^{2}=\frac{19^{2}+22^{2}}{5}=13^{2}\)

\( \Rightarrow \sqrt{4a^{2}+4b^{2}}=XY=\sqrt{4× 169}=\boxed{26}\) – Aneesh Kundu · 2 years ago

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Length of XY is 26 – Moumik Maitra · 1 year, 10 months ago

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XY = 26 – KahYeen Lai · 1 year, 10 months ago

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XY=26 – KahYeen Lai · 1 year, 10 months ago

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26 – Unity 002 · 1 year, 10 months ago

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26 – Sudharshan Sharma · 1 year, 10 months ago

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