2018 AMC 10A Problems/Problem 16
Contents
Problem
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
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)
Solution 2 - Circles
Note that if a circle with an integer radius centered at vertex
intersects hypotenuse
, the lines drawn from
to the points of intersection are integer lengths. As in the previous solution, the shortest distance
. As a result, a circle of
will [b]not[/b] reach the hypotenuse and thus does not intersect it. We also know that a circle of radius
intersects the hypotenuse once and a circle of radius
intersects the hypotenuse twice. Quick graphical thinking or Euclidean construction will prove this.
unitsize(4); 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("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, NE); dot("$P$", P, S);] draw(circle((0,0), 21)); draw(circle((0,0), 20)); draw(circle((0,0), 19)); draw(circle((0,0), 18)); draw(circle((0,0), 17)); draw(circle((0,0), 16)); draw(circle((0,0), 15)); (Error making remote request. Unknown error_msg)
It follows that we can draw circles of radii and
that each contribute [b]two[/b] integer lengths from
to
and one circle of radius
that contributes only one such segment. Our answer is then
~samrocksnature
Video Solution 1
~IceMatrix
Video Solution 2
https://youtu.be/4_x1sgcQCp4?t=3790
~ pi_is_3.14
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AMC 10 Problems and Solutions |
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