Difference between revisions of "2019 AIME I Problems/Problem 3"
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Knowing that <math>\triangle{PQR}</math> has area <math>150</math> and is a <math>3</math>-<math>4</math>-<math>5</math> triangle, we can find the area of the smaller triangles <math>\triangle{DRE}</math>, <math>\triangle{APF}</math>, and <math>\triangle{CQB}</math> and subtract them from <math>\triangle{PQR}</math> to obtain our answer. First off, we know <math>\triangle{DRE}</math> has area <math>12.5</math> since it is a right triangle. To the find the areas of <math>\triangle{APF}</math> and <math>\triangle{CQB}</math> , we can use Law of Cosines (<math>c^2 = a^2 + b^2 - 2ab\cos C</math>) to find the lengths of <math>AF</math> and <math>CB</math>, respectively. Computing gives <math>AF = \sqrt{20}</math> and <math>CB = \sqrt{10}</math>. Now, using Heron's Formula, we find <math>\triangle{APF} = 10</math> and <math>\triangle{CQB} = 7.5</math>. Adding these and subtracting from <math>\triangle{PQR}</math>, we get <math>150 - (10 + 7.5 + 12.5) = \boxed{120}</math> -Starsher | Knowing that <math>\triangle{PQR}</math> has area <math>150</math> and is a <math>3</math>-<math>4</math>-<math>5</math> triangle, we can find the area of the smaller triangles <math>\triangle{DRE}</math>, <math>\triangle{APF}</math>, and <math>\triangle{CQB}</math> and subtract them from <math>\triangle{PQR}</math> to obtain our answer. First off, we know <math>\triangle{DRE}</math> has area <math>12.5</math> since it is a right triangle. To the find the areas of <math>\triangle{APF}</math> and <math>\triangle{CQB}</math> , we can use Law of Cosines (<math>c^2 = a^2 + b^2 - 2ab\cos C</math>) to find the lengths of <math>AF</math> and <math>CB</math>, respectively. Computing gives <math>AF = \sqrt{20}</math> and <math>CB = \sqrt{10}</math>. Now, using Heron's Formula, we find <math>\triangle{APF} = 10</math> and <math>\triangle{CQB} = 7.5</math>. Adding these and subtracting from <math>\triangle{PQR}</math>, we get <math>150 - (10 + 7.5 + 12.5) = \boxed{120}</math> -Starsher | ||
+ | ==Solution 5 (Official MAA)== | ||
+ | Triangle <math>PQR</math> is a right triangle with are <math>\tfrac12\cdot15\cdot20=150</math>. Each of <math>\triangle PAF,\triangle QCB,</math> and <math>\triangle RED</math> shares an angle with <math>\triangle PQR</math>. Because the area of a triangle with sides <math>a,\,b,</math> and included angle <math>\gamma</math> is <math>\tfrac12a\cdot b\cdot \sin\gamma,</math> it follows that the areas of <math>\triangle PAF,\triangle QCB,</math> and <math>\triangle RED</math> are each <math>\tfrac12\cdot 5\cdot5\cdot\tfrac{150}{ab},</math> where <math>a</math> and <math>b</math> are the lengths of the sides of <math>\triangle PQR</math> adjacent to the shared angle. Thus the sum of the areas of <math>\triangle PAF,\triangle QCB,</math> and <math>\triangle RED</math> is <cmath>5\cdot5\cdot\frac{150}{15\cdot25}+5\cdot5\cdot\frac{150}{25\cdot20}+5\cdot5\cdot\frac{150}{20\cdot15}=25\left(\frac25+\frac3{10}+\frac12\right)=30.</cmath> Therefore <math>ABCDEF</math> has area <math>150-30=120</math>. | ||
==Video Solution #1(Complementary Area Counting?)== | ==Video Solution #1(Complementary Area Counting?)== | ||
https://youtu.be/JQdad7APQG8?t=417 | https://youtu.be/JQdad7APQG8?t=417 |
Revision as of 15:39, 25 February 2021
Contents
Problem
In ,
,
, and
. Points
and
lie on
, points
and
lie on
, and points
and
lie on
, with
. Find the area of hexagon
.
Diagram
Solution 1
We know the area of the hexagon to be
. Since
, we know that
is a right triangle. Thus the area of
is
. Another way to compute the area is
Then the area of
. Preceding in a similar fashion for
, the area of
is
. Since
, the area of
. Thus our desired answer is
Solution 2
Let be the origin. Noticing that the triangle is a 3-4-5 right triangle, we can see that
, and
. Using the shoelace theorem, the area is
.
Shoelace theorem:Suppose the polygon
has vertices
,
, ... ,
, listed in clockwise order. Then the area of
is
You can also go counterclockwise order, as long as you find the absolute value of the answer.
.
Solution 3 (Easiest, uses only basic geometry too)
Note that has area
and is a
-
-
right triangle. Then, by similar triangles, the altitude from
to
has length
and the altitude from
to
has length
, so
, meaning that
.
-Stormersyle
Solution 4
Knowing that has area
and is a
-
-
triangle, we can find the area of the smaller triangles
,
, and
and subtract them from
to obtain our answer. First off, we know
has area
since it is a right triangle. To the find the areas of
and
, we can use Law of Cosines (
) to find the lengths of
and
, respectively. Computing gives
and
. Now, using Heron's Formula, we find
and
. Adding these and subtracting from
, we get
-Starsher
Solution 5 (Official MAA)
Triangle is a right triangle with are
. Each of
and
shares an angle with
. Because the area of a triangle with sides
and included angle
is
it follows that the areas of
and
are each
where
and
are the lengths of the sides of
adjacent to the shared angle. Thus the sum of the areas of
and
is
Therefore
has area
.
Video Solution #1(Complementary Area Counting?)
https://youtu.be/JQdad7APQG8?t=417
Video Solution
https://www.youtube.com/watch?v=4jOfXNiQ6WM
Video Solution 2
https://youtu.be/TSKcjht8Rfk?t=941
~IceMatrix
Video Solution 3
~Shreyas S
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.