2011 AMC 10B Problems/Problem 25
Problem
Let be a triangle with sides
and
. For
, if
and
and
are the points of tangency of the incircle of
to the sides
and
respectively, then
is a triangle with side lengths
and
if it exists. What is the perimeter of the last triangle in the sequence
?
Solution
By constructing the bisectors of each angle and the perpendicular radii of the incircle the triangle consists of 3 kites. Hence and
and
. Let
and
gives three equations:
(where for the first triangle.)
Solving gives:
Subbing in gives that has sides of
.
Repeating gives with sides
.
has sides
.
has sides
.
has sides
.
has sides
.
has sides
.
has sides
.
has sides
.
would have sides
but these length do not make a
triangle as
.
Hence the perimeter is
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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All AMC 10 Problems and Solutions |