2002 AMC 12B Problem 20

Problem 20 of 25HarderGeometry

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.

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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.

More practice

Concepts: right triangle · median (geometry) · Pythagorean Theorem

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.