2006 AMC 12B Problems/Problem 23
Contents
Problem
Isosceles has a right angle at
. Point
is inside
, such that
,
, and
. Legs
and
have length $s=\sqrt{a+b\sqrt{2}{$ (Error compiling LaTeX. Unknown error_msg), where
and
are positive integers. What is
?
Solution
Using the Law of Cosines on
, we have:
Using the Law of Cosines on , we have:
Now we use .
Note that we know that we want the solution with since we know that
. Thus,
.
Solution 2
Rotate triangle 90 degrees counterclockwise about
so that the image of
rests on
. Now let the image of
be
. Note that
, meaning triangle
is right isosceles, and
. Then
. Now because
and
, we observe that
, by the Pythagorean Theorem on
. Now we have that
. So we take the cosine of the second equality, using that fact that
, to get
. Finally, we use the fact that
and use the Law of Cosines on triangle
to arrive at the value of
.
See also
2006 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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 |