Difference between revisions of "1989 AIME Problems/Problem 15"
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Firstly, since they all meet at one single point, denoting the mass of them separately. Assuming <math>M(A)=6;M(D)=6;M(B)=3;M(E)=9</math>; we can get that <math>M(P)=12;M(F)=9;M(C)=3</math>; which leads to the ratio between segments, <math>\frac{CE}{AE}=2;\frac{BF}{AF}=2;\frac{BD}{CD}=1</math>. Denoting that <math>CE=2x;AE=x; AF=y; BF=2y; CD=z; DB=z.</math> | Firstly, since they all meet at one single point, denoting the mass of them separately. Assuming <math>M(A)=6;M(D)=6;M(B)=3;M(E)=9</math>; we can get that <math>M(P)=12;M(F)=9;M(C)=3</math>; which leads to the ratio between segments, <math>\frac{CE}{AE}=2;\frac{BF}{AF}=2;\frac{BD}{CD}=1</math>. Denoting that <math>CE=2x;AE=x; AF=y; BF=2y; CD=z; DB=z.</math> | ||
Now we know three cevians' length, Applying Stewart theorem to them, getting three different equations: | Now we know three cevians' length, Applying Stewart theorem to them, getting three different equations: | ||
+ | |||
<math>(1): (3x)^2 * 2y+(2z)^2 * y=(3y)(2y^2+400)\\ | <math>(1): (3x)^2 * 2y+(2z)^2 * y=(3y)(2y^2+400)\\ | ||
+ | |||
(2): (3y)^2 * z+(3x)^2 * z=(2z)(z^2+144)\\ | (2): (3y)^2 * z+(3x)^2 * z=(2z)(z^2+144)\\ | ||
+ | |||
(3): (2z)^2 * x+(3y)^2 * x=(3x)(2x^2+144)</math> | (3): (2z)^2 * x+(3y)^2 * x=(3x)(2x^2+144)</math> | ||
+ | |||
After solving the system of equation, we get that <math>x=3\sqrt{5};y=\sqrt{13};z=3\sqrt{13}</math>; | After solving the system of equation, we get that <math>x=3\sqrt{5};y=\sqrt{13};z=3\sqrt{13}</math>; | ||
+ | |||
pulling <math>x,y,z</math> back to get the length of <math>AC=9\sqrt{5};AB=3\sqrt{13};BC=6\sqrt{13}</math>; now we can apply Heron's formula here, which is <math>\sqrt\frac{(9\sqrt{5}+9\sqrt{13})(9\sqrt{13}-9\sqrt{5})(9\sqrt{5}+3\sqrt{13})(9\sqrt{5}-3\sqrt{13})}{16}=108</math> | pulling <math>x,y,z</math> back to get the length of <math>AC=9\sqrt{5};AB=3\sqrt{13};BC=6\sqrt{13}</math>; now we can apply Heron's formula here, which is <math>\sqrt\frac{(9\sqrt{5}+9\sqrt{13})(9\sqrt{13}-9\sqrt{5})(9\sqrt{5}+3\sqrt{13})(9\sqrt{5}-3\sqrt{13})}{16}=108</math> | ||
+ | |||
Our answer is <math>\boxed{108}</math> | Our answer is <math>\boxed{108}</math> | ||
~bluesoul | ~bluesoul |
Revision as of 03:58, 23 December 2021
Contents
Problem
Point is inside
. Line segments
,
, and
are drawn with
on
,
on
, and
on
(see the figure below). Given that
,
,
,
, and
, find the area of
.
Solution
Solution 1
Because we're given three concurrent cevians and their lengths, it seems very tempting to apply Mass points. We immediately see that ,
, and
. Now, we recall that the masses on the three sides of the triangle must be balanced out, so
and
. Thus,
and
.
Recalling that , we see that
and
is a median to
in
. Applying Stewart's Theorem,
, and
. Now notice that
, because both triangles share the same base and the
. Applying Heron's formula on triangle
with sides
,
, and
,
and
.
Solution 2
Using a different form of Ceva's Theorem, we have
Solving and
, we obtain
and
.
Let be the point on
such that
.
Since
and
,
. (Stewart's Theorem)
Also, since and
, we see that
,
, etc. (Stewart's Theorem)
Similarly, we have
(
) and thus
.
is a
right triangle, so
(
) is
.
Therefore, the area of
.
Using area ratio,
.
Solution 3
Because the length of cevian is unknown, we can examine what happens when we extend it or decrease its length and see that it simply changes the angles between the cevians. Wouldn't it be great if it the length of
was such that
? Let's first assume it's a right angle and hope that everything works out.
Extend to
so that
. The result is that
,
, and
because
. Now we see that if we are able to show that
, that is
, then our right angle assumption will be true.
Apply the Pythagorean Theorem on to get
, so
and
. Now, we apply the Law of Cosines on triangles
and
.
Let . Notice that
and
, so we get two nice equations.
Solving, (yay!).
Now, the area is easy to find. .
[however, I think this solution is wrong, the A, B, and Cs do not match with the picture]
Solution 4
First, let and
Thus, we can easily find that
Now,
In the same manner, we find that
Now, we can find that
We can now use this to find that
Plugging this value in, we find that
Now, since
we can find that
Setting
we can apply Stewart's Theorem on triangle
to find that
Solving, we find that
But,
meaning that
Since
we conclude that the answer is
.
Solution 5(Mass of a point+ Stewart+heron)
Firstly, since they all meet at one single point, denoting the mass of them separately. Assuming ; we can get that
; which leads to the ratio between segments,
. Denoting that
Now we know three cevians' length, Applying Stewart theorem to them, getting three different equations:
$(1): (3x)^2 * 2y+(2z)^2 * y=(3y)(2y^2+400)\\
(2): (3y)^2 * z+(3x)^2 * z=(2z)(z^2+144)\\
(3): (2z)^2 * x+(3y)^2 * x=(3x)(2x^2+144)$ (Error compiling LaTeX. Unknown error_msg)
After solving the system of equation, we get that ;
pulling back to get the length of
; now we can apply Heron's formula here, which is
Our answer is
~bluesoul
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Final Question | |
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.