Let \triangle XOY be a right-angled triangle with m\angle XOY=90°. Let M and N be the midpoints of legs OX and OY, respectively. Given that XN=19 and YM=22, find XY.

2002 AMC 12B Problem 20 — Solution

2002 AMC 12B Problem 20

Below is the professionally curated solution for Problem 20 of the 2002 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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Concepts:right triangle median (geometry)Pythagorean Theorem

Difficulty rating: 1660

20.

Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY=90^\circ.$ Let $M$ and $N$ be the midpoints of legs $OX$ and $OY,$ respectively. Given that $XN=19$ and $YM=22,$ find $XY.$

$24$

$26$

$28$

$30$

$32$

Solution:

Let $OM=a$ and $ON=b.$ Since $M,N$ are midpoints, $XN^2=(2a)^2+b^2=361$ and $YM^2=a^2+(2b)^2=484.$

Adding, $5(a^2+b^2)=845,$ so $a^2+b^2=169.$ Then $MN=\sqrt{a^2+b^2}=13,$ and since $MN$ is the midsegment joining $M$ and $N,$ we have $XY=2\,MN=26.$

Thus, the correct answer is B.

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