2004 AMC 10B Problems/Problem 24
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In triangle we have
,
,
. Point
is on the circumscribed circle of the triangle so that
bisects angle
. What is the value of
?
Solution 1
Set 's length as
.
's length must also be
since
and
intercept arcs of equal length(because
). Using Ptolemy's Theorem,
. The ratio is
Solution 2
Let
. Observe that
because they subtend the same arc. Furthermore,
, so
is similar to
by AAA similarity. Then
. By angle bisector theorem,
so
which gives
. Plugging this into the similarity proportion gives:
.
See Also
2004 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 |
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