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

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestis XY 26????

Log in to reply

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

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

\( \Rightarrow \sqrt{4a^{2}+4b^{2}}=XY=\sqrt{4× 169}=\boxed{26}\)

Log in to reply

Length of XY is 26

Log in to reply

XY = 26

Log in to reply

XY=26

Log in to reply

26

Log in to reply

26

Log in to reply