2021 Fall AMC 12A Problems/Problem 13
Contents
Problem
The angle bisector of the acute angle formed at the origin by the graphs of the lines and
has equation
What is
Diagram
Solution 1
IN PROGRESS. APPRECIATE IT IF NO EDITS. THANKS.
~MRENTHUSIASM
Solution 2
Note that the distance between the point to line
is
Because line
is a perpendicular bisector, a point on the line
must be equidistant from the two lines(
and
), call this point
Because, the line
passes through the origin, our requested value of
which is the slope of the angle bisector line, can be found when evaluating the value of
By the Distance from Point to Line formula we get the equation,
Note that
because
is higher than
and
because
is lower to
Thus, we solve the equation,
Thus, the value of
Thus, the answer is
(Fun Fact: The value is the golden ratio
)
~NH14
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 |
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