2018 AMC 10A Problems/Problem 16
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
[asy]
pair A, B, C, E, P;
A=(-20, 0);
B=origin;
C=(0,21);
E=(-21, 20);
P=extension(B,E, A, C);
draw(A--B--C--cycle);
draw(B--P);
dot("
", A, SW);
dot("", B, SE);
dot("
", C, NE);
dot("
", P, S);
[/asy]
As the problem has no diagram, we draw a diagram. 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 
(IVT). Similarly, moving from to hits all the integer values from 
. This is a total of 
line segments.
(asymptote diagram added by elements2015)
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 | ||
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