Difference between revisions of "2021 AIME I Problems/Problem 2"
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By the Pythagorean theorem again, <math>{CG}^2={EG}^2+{CE}^2</math>: <cmath>11-x=\sqrt{7^2+(9-\sqrt{x^2+9})^2}.</cmath> | By the Pythagorean theorem again, <math>{CG}^2={EG}^2+{CE}^2</math>: <cmath>11-x=\sqrt{7^2+(9-\sqrt{x^2+9})^2}.</cmath> | ||
− | Solving, we get <math>x=\frac{9}{4}</math>, so the area of the parallelogram is <math>3\cdot(11-\frac{9}{4})=\frac{105}{4}</math>, and <math>105+4=\framebox{109}</math>. | + | Solving, we get <math>x=\frac{9}{4}</math>, so the area of the parallelogram is <math>3\cdot\left(11-\frac{9}{4}\right)=\frac{105}{4}</math>, and <math>105+4=\framebox{109}</math>. |
~JulianaL25 | ~JulianaL25 |
Revision as of 01:49, 20 December 2021
Contents
- 1 Problem
- 2 Solution 1 (Similar Triangles)
- 3 Solution 2 (Pythagorean Theorem)
- 4 Solution 3 (Similar Triangles and Area)
- 5 Solution 4
- 6 Solution 5 (Coordinate Geometry Bash)
- 7 Solution 6 (Trigonometry Bash)
- 8 Video Solution by Punxsutawney Phil
- 9 Video Solution
- 10 Video Solution by Steven Chen (in Chinese)
- 11 Video Solution
- 12 See Also
Problem
In the diagram below, is a rectangle with side lengths
and
, and
is a rectangle with side lengths
and
as shown. The area of the shaded region common to the interiors of both rectangles is
, where
and
are relatively prime positive integers. Find
.
Solution 1 (Similar Triangles)
Let be the intersection of
and
.
From vertical angles, we know that
. Also, because we are given that
and
are rectangles, we know that
.
Therefore, by AA similarity, we know that
.
Let . Then, we have
. By similar triangles, we know that
and
. We have
.
Solving for , we have
.
The area of the shaded region is just
.
Thus, the answer is .
~yuanyuanC
Solution 2 (Pythagorean Theorem)
Let the intersection of and
be
, and let
, so
.
By the Pythagorean theorem, , so
, and thus
.
By the Pythagorean theorem again, :
Solving, we get , so the area of the parallelogram is
, and
.
~JulianaL25
Solution 3 (Similar Triangles and Area)
Again, let the intersection of and
be
. By AA similarity,
with a
ratio. Define
as
. Because of similar triangles,
. Using
, the area of the parallelogram is
. Using
, the area of the parallelogram is
. These equations are equal, so we can solve for
and obtain
. Thus,
, so the area of the parallelogram is
.
Finally, the answer is .
~mathboy100
Solution 4
Let , and
. Also let
.
also has to be
by parallelogram properties. Then
and
must be
because the sum of the segments has to be
.
We can easily solve for by the Pythagorean Theorem:
It follows shortly that
.
Also, , and
. We can then say that
, so
.
Now we can apply the Pythagorean Theorem to .
This simplifies (not-as-shortly) to . Now we have to solve for the area of
. We know that the height is
because the height of the parallelogram is the same as the height of the smaller rectangle.
From the area of a parallelogram (we know that the base is and the height is
), it is clear that the area is
, giving an answer of
.
~ishanvannadil2008 (Solution Sketch)
~Tuatara (Rephrasing and )
Solution 5 (Coordinate Geometry Bash)
Suppose It follows that
and
Since is a rectangle, we have
and
The equation of the circle with center
and radius
is
and the equation of the circle with center
and radius
is
We now have a system of two equations with two variables. Expanding and rearranging respectively give
Subtracting
from
we obtain
Simplifying and rearranging produce
Substituting
into
gives
which is a quadratic of
We clear fractions by multiplying both sides by
then solve by factoring:
Since
is in Quadrant IV, we have
It follows that the equation of
is
Let be the intersection of
and
and
be the intersection of
and
Since
is the
-intercept of
we get
By symmetry, quadrilateral is a parallelogram. Its area is
from which the requested sum is
~MRENTHUSIASM
Solution 6 (Trigonometry Bash)
Let the intersection of and
be
. It is useful to find
, because
and
. From there, subtracting the areas of the two triangles from the larger rectangle, we get Area =
.
let . Let
. Note,
.
. The answer is
.
~twotothetenthis1024
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=H17E9n2nIyY&t=289s
Video Solution
https://youtu.be/M3DsERqhiDk?t=275
Video Solution by Steven Chen (in Chinese)
Video Solution
https://www.youtube.com/watch?v=BinfKrc5bWo
See Also
2021 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.