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 | |
| 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 10 Problems and Solutions | ||
