2019 AIME I Problems/Problem 6
In convex quadrilateral side is perpendicular to diagonal , side is perpendicular to diagonal , , and . The line through perpendicular to side intersects diagonal at with . Find .
Solution 1 (Trig)
Let and . Note .
Then, . Furthermore, .
Dividing the equations gives
Thus, , so .
Solution 2 (Similar triangles)
First, let be the intersection of and as shown above. Note that as given in the problem. Since and , by AA similarity. Similarly, . Using these similarities we see that and Combining the two equations, we get Since , we get .
Solution by vedadehhc
Solution 3 (Similar triangles, orthocenters)
Extend and past and respectively to meet at . Let be the intersection of diagonals and (this is the orthocenter of ).
As (as , using the fact that is the orthocenter), we may let and .
Then using similarity with triangles and we have
Cross-multiplying and dividing by gives so . (Solution by scrabbler94)
Solution 4 (Algebraic Bashing)
First, let be the intersection of and . We can use the right triangles in the problem to create equations. Let and We are trying to find We can find equations. They are and We can subtract the fifth equation from the sixth equation to get We can subtract the fourth equation from the third equation to get Combining these equations gives so Substituting this into the seventh equation gives Substituting this into the second equation gives . Subtracting the first equation from this gives Solving this equation, we find that (Solution by DottedCaculator)
Solution 5 (5-second PoP)
Notice that is inscribed in the circle with diameter and is inscribed in the circle with diameter . Furthermore, is tangent to . Then, and .
(Solution by TheUltimate123)
If you're wondering why it's because $KX\cdotKN=KX\cdot(KX+XN)=KX^2+KX\cdotKN=KX^2+LX^2=KL^2$ (Error compiling LaTeX. ! Undefined control sequence.) (last part by similarity).
Solution 6 (Alternative PoP)
(Diagram by vedadehhc)
Call the base of the altitude from to point . Let . Now, we have that by the Pythagorean Theorem. Once again by Pythagorean, . Using Power of a Point, we have
( is the intersection of with the circle )
(Solution by RootThreeOverTwo)
Solution 7 (just one pair of similar triangles)
Note that since , quadrilateral is cyclic. Therefore, we have so , giving Therefore, .
Video Solution: https://www.youtube.com/watch?v=0AXF-5SsLc8
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