Let △ XOY be a right-angled triangle with m∠ 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

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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

Let

OM=a,

ON=b

; then

19^2=(2a)^2+b^2

and

22^2=a^2+(2b)^2

Add the equations to get

5(a^2+b^2)

; note that

XY=2\sqrt{a^2+b^2}

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,

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.