2021 Fall AMC 12B Problems/Problem 10
Contents
Problem
What is the sum of all possible values of between
and
such that the triangle in the coordinate plane whose vertices are
is isosceles?
Solution 1
Let and
We apply casework to the legs of isosceles
Note that
must be the midpoint of
It follows that
so
Note that
must be the midpoint of
It follows that
so
Note that
must be the midpoint of
It follows that
or
so
or
Together, the sum of all such possible values of is
Remark
The following diagram shows all possible locations of
Solution 2 Cheese 🧀
It is fairly unlikely that the sum of all angles is less than 360, therefore we choose 380 as our answer.
vids
~Steven Chen (www.professorchenedu.com) ~Wilhelm Z ~MRENTHUSIASM
Video Solution (Just 1 min!)
~Education, the Study of Everything
Video Solution by TheBeautyofMath
https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1304
~IceMatrix
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 12 Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.