Difference between revisions of "2014 AIME I Problems/Problem 15"
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===Solution 3=== | ===Solution 3=== | ||
Call <math>DE=x</math> and as a result <math>DG=EF=\frac{x\sqrt{2}}{2}, EF=\frac{3x}{5}, FD=\frac{4x}{5}</math>. Since <math>EFGD</math> is cyclic we just need to get <math>DG</math> and using LoS(for more detail see the <math>2</math>nd paragraph of Solution <math>2</math>) we get <math>AG=\frac{5}{2}</math> and using a similar argument(use LoS again) and subtracting you get <math>FG=\frac{5}{14}</math> so you can use Ptolemy to get <math>x=\frac{25\sqrt{2}}{14} \implies \boxed{041}</math>. | Call <math>DE=x</math> and as a result <math>DG=EF=\frac{x\sqrt{2}}{2}, EF=\frac{3x}{5}, FD=\frac{4x}{5}</math>. Since <math>EFGD</math> is cyclic we just need to get <math>DG</math> and using LoS(for more detail see the <math>2</math>nd paragraph of Solution <math>2</math>) we get <math>AG=\frac{5}{2}</math> and using a similar argument(use LoS again) and subtracting you get <math>FG=\frac{5}{14}</math> so you can use Ptolemy to get <math>x=\frac{25\sqrt{2}}{14} \implies \boxed{041}</math>. | ||
+ | ~First | ||
== See also == | == See also == | ||
{{AIME box|year=2014|n=I|num-b=14|after=Last Question}} | {{AIME box|year=2014|n=I|num-b=14|after=Last Question}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:08, 25 June 2017
Problem 15
In ,
,
, and
. Circle
intersects
at
and
,
at
and
, and
at
and
. Given that
and
, length
, where
and
are relatively prime positive integers, and
is a positive integer not divisible by the square of any prime. Find
.
Solution 1
Since ,
is the diameter of
. Then
. But
, so
is a 45-45-90 triangle. Letting
, we have that
,
, and
.
Note that by SAS similarity, so
and
. Since
is a cyclic quadrilateral,
and
, implying that
and
are isosceles. As a result,
, so
and
.
Finally, using the Pythagorean Theorem on ,
Solving for
, we get that
, so
. Thus, the answer is
.
Solution 2
First we note that is an isosceles right triangle with hypotenuse
the same as the diameter of
. We also note that
since
is a right angle and the ratios of the sides are
.
From congruent arc intersections, we know that , and that from similar triangles
is also congruent to
. Thus,
is an isosceles triangle with
, so
is the midpoint of
and
. Similarly, we can find from angle chasing that
. Therefore,
is the angle bisector of
. From the angle bisector theorem, we have
, so
and
.
Lastly, we apply power of a point from points and
with respect to
and have
and
, so we can compute that
and
. From the Pythagorean Theorem, we result in
, so
Also: . We can also use Ptolemy's Theorem on quadrilateral
to figure what
is in terms of
:
Thus
.
Solution 3
Call and as a result
. Since
is cyclic we just need to get
and using LoS(for more detail see the
nd paragraph of Solution
) we get
and using a similar argument(use LoS again) and subtracting you get
so you can use Ptolemy to get
.
~First
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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