Pre-RMO 2014/15

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

#Geometry #Pre-RMO

Note by Pranshu Gaba
6 years, 8 months ago

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Comments

is XY 26????

Nitish Deshpande - 6 years, 8 months ago

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Yes .XY=26 is the correct answer.

D K - 2 years, 10 months ago

Let XM=MO=aXM=MO=a and ON=NY=bON=NY=b

By applying PT, we get

4a2+b2=192→Eq.14a^{2}+b^{2}=19^{2} \rightarrow Eq.1

a2+4b2=222→Eq.2a^{2}+4b^{2}=22^{2} \rightarrow Eq.2

Eq.1+Eq.2

a2+b2=192+2225=132a^{2}+b^{2}=\frac{19^{2}+22^{2}}{5}=13^{2}

⇒4a2+4b2=XY=4×169=26 \Rightarrow \sqrt{4a^{2}+4b^{2}}=XY=\sqrt{4× 169}=\boxed{26}

Aneesh Kundu - 6 years, 8 months ago

26

sudharshan sharma - 6 years, 6 months ago

26

unity 002 - 6 years, 6 months ago

XY=26

KahYeen Lai - 6 years, 6 months ago

XY = 26

KahYeen Lai - 6 years, 6 months ago

Length of XY is 26

Moumik Maitra - 6 years, 6 months ago
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