2024 AMC 12B Problems/Problem 20
Contents
Problem 20
Suppose ,
, and
are points in the plane with
and
, and let
be the length of the line segment from
to the midpoint of
. Define a function
by letting
be the area of
. Then the domain of
is an open interval
, and the maximum value
of
occurs at
. What is
?
Solution
Let the midpoint of be
, and let the length
. We know there are limits to the value of
, and these limits can probably be found through the triangle inequality. But the triangle inequality relates the third side length
to
and
, and doesn't contain any information about the median. Therefore we're going to have to write the side
in terms of
and then use the triangle inequality to find bounds on
.
We use Stewart's theorem to relate to the median
:
. In this case
,
,
,
,
,
.
Therefore we get the equation
.
Notice that since is a pythagorean triple, this means
.
By triangle inequality, and
Let's tackle the first inequality:
Solution #1
Let midpoint of as
, extends
to
and
,
triangle has
sides
as such,
so
so
which is achieved when
, then
See also
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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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