2025 AIME I Problems/Problem 2
Contents
[hide]Problem
On points
,
,
, and
lie in that order on side
with
,
, and
. Points
,
,
, and
lie in that order on side
with
,
, and
. Let
be the reflection of
through
, and let
be the reflection of
through
. Quadrilateral
has area
. Find the area of heptagon
.
Solution 1
Note that the triangles outside have the same height as the unshaded triangles in
. Since they have the same bases, the area of the heptagon is the same as the area of triangle
. Therefore, we need to calculate the area of
. Denote the length of
as
and the altitude of
to
as
. Since
,
and the altitude of
is
. The area
. The area of
is equal to
.
~alwaysgonnagiveyouup
Solution 2
Because of reflections, and various triangles having the same bases, we can conclude that . Through the given lengths of
on the left and
on the right, we conclude that the lines through
are parallel, and the sides are in a
ratio. Because these lines are parallel, we can see that
, are similar, and from our earlier ratio, we can give the triangles side ratios of
, or area ratios of
. Quadrilateral
corresponds to the
, which corresponds to the ratio
. Dividing
by
, we get
, and finally multiplying
gives us our answer of
~shreyan.chethan, cleaned up by cweu001
Solution 3
By area lemma, we can see that the areas of the shaded areas are equivalent to the areas of the unshaded areas. Thus, we see that the desired area is equivalent to the area of the triangle . Since
, we have
, meaning
. Thus, since
, we can calculate
.
~cweu001, cleaned up by shreyan.chethan
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=J-0BapU4Yuk
Video Solution(Fast! Easy!)
~MC
Video Solution by Mathletes Corner
https://www.youtube.com/watch?v=fVBk2vOusio&t=3s
~GP102
See also
2025 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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