Difference between revisions of "2018 AMC 10A Problems/Problem 16"
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https://youtu.be/M22S82Am2zM | https://youtu.be/M22S82Am2zM | ||
~IceMatrix | ~IceMatrix | ||
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+ | == Video Solution 2 == | ||
+ | https://youtu.be/4_x1sgcQCp4?t=3790 | ||
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+ | ~ pi_is_3.14 | ||
==See Also== | ==See Also== |
Revision as of 20:40, 24 January 2021
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)
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 | |
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.