Difference between revisions of "2021 Fall AMC 12B Problems/Problem 10"
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~Steven Chen (www.professorchenedu.com) ~Wilhelm Z ~MRENTHUSIASM | ~Steven Chen (www.professorchenedu.com) ~Wilhelm Z ~MRENTHUSIASM | ||
− | == Video Solution | + | |
− | https://youtu.be/4qgYrCYG-qw | + | |
+ | ==Video Solution (Just 1 min!)== | ||
+ | https://youtu.be/F_Hy5OWBC54 | ||
+ | |||
+ | <i>~Education, the Study of Everything</i> | ||
+ | |||
+ | == Video Solution by TheBeautyofMath == | ||
+ | https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1304 | ||
+ | |||
+ | ~IceMatrix | ||
== See Also == | == See Also == | ||
{{AMC12 box|year=2021 Fall|ab=B|num-a=11|num-b=9}} | {{AMC12 box|year=2021 Fall|ab=B|num-a=11|num-b=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:50, 10 July 2023
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
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
~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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.