Difference between revisions of "2025 AIME II Problems/Problem 14"
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| + | Let <math>{\triangle ABC}</math> be a right triangle with <math>\angle A = 90^\circ</math> and <math>BC = 38.</math> There exist points <math>K</math> and <math>L</math> inside the triangle such<cmath>AK = AL = BK = CL = KL = 14.</cmath>The area of the quadrilateral <math>BKLC</math> can be expressed as <math>n\sqrt3</math> for some positive integer <math>n.</math> Find <math>n.</math> | ||
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==Solution 1(Coordinates and Bashy Algebra)== | ==Solution 1(Coordinates and Bashy Algebra)== | ||
Revision as of 21:55, 13 February 2025
Let 
be a right triangle with 
and 
There exist points 
and 
inside the triangle such![\[AK = AL = BK = CL = KL = 14.\]](http://latex.artofproblemsolving.com/c/b/3/cb31cd53c432faa251c220830b62d5a30ee8f6df.png)
The area of the quadrilateral 
can be expressed as 
for some positive integer 
Find
Solution 1(Coordinates and Bashy Algebra)
By drawing our the triangle, I set A to be (0, 0) in the coordinate plane. I set C to be (x, 0) and B to be (0, y). I set K to be (a, b) and L to be (c, d). Then, since all of these distances are 14, I used coordinate geometry to set up the following equations:
+ 
= 196; + 
= 196; 
+ 
= 196; 
+ 
= 196; 
+ . = 196. Notice by merging the first two equations, the only possible way for it to work is if 
= 
which means 
. Next, since the triangle is right, and we know one leg is 
as , the other leg, x, is 
.Then, plugging these in, we get a system of equations with 4 variables and 4 equations and solving, we get a = 2, b = 8
, c = 13, d = 3. Now plugging in all the points and using the Pythagorean Theorem, we get the coordinates of the quadrilateral. By Shoelace, our area is 104. Thus, the answer is 
.
~ilikemath247365
