Difference between revisions of "2018 AMC 10A Problems/Problem 16"
m (→Solution) |
m (→See Also) |
||
Line 17: | Line 17: | ||
==See Also== | ==See Also== | ||
− | {{AMC10 box|year=2018|ab=A|num-b= | + | {{AMC10 box|year=2018|ab=A|num-b=15|num-a=17}} |
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 17:38, 8 February 2018
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.
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.