2018 AMC 10A Problems/Problem 16
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Right triangle has leg lengths
and
. Including
and
, how many line segments with integer length can be drawn from vertex
to a point on hypotenuse
?
Solution
The hypotenuse has length . Let
be the foot of the altitude from
to
. Note that
is the shortest possible length of any segment. Writing the area of the triangle in two ways, we can solve for
, which is between
and
.
Let the line segment be , with
on
. As you move
along the hypotenuse from
to
, the length of
strictly decreases, hitting all the integer values from
. Similarly, moving
from
to
hits all the integer values from
. This is a total of
line segments.