Difference between revisions of "2019 AIME I Problems/Problem 6"
Scrabbler94 (talk | contribs) (→Solution (Similar triangles): add alternate solution) |
(→Solution (Similar triangles)) |
||
Line 2: | Line 2: | ||
In convex quadrilateral <math>KLMN</math> side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | In convex quadrilateral <math>KLMN</math> side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | ||
+ | ==Solution 1== | ||
+ | Let <math>\angle MKN=\alpha</math> and <math>\angle LNK=\beta</math>. Note <math>\angle KLP=\beta</math>. | ||
+ | |||
+ | Then, <math>KP=28\sin\beta=8\cos\alpha</math>. | ||
+ | Furthermore, <math>KN=65/\sin\alpha=28/\sin\beta \Rightarrow 65\sin\beta=28\sin\alpha</math>. | ||
+ | |||
+ | Dividing the equations gives | ||
+ | <cmath>\frac{7}{2}\tan\alpha=\frac{65}{28}\Rightarrow \tan\alpha=\frac{65}{98}</cmath> | ||
+ | |||
+ | Thus, <math>MK=\frac{MN}{\tan\alpha}=98</math>, so <math>MO=MK-KO=\boxed{90}</math>. | ||
==Solution (Similar triangles)== | ==Solution (Similar triangles)== | ||
(writing this, don't edit) | (writing this, don't edit) |
Revision as of 13:01, 15 March 2019
Contents
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
Let and . Note .
Then, . Furthermore, .
Dividing the equations gives
Thus, , so .
Solution (Similar triangles)
(writing this, don't edit)
First, let be the intersection of and . Note that as given in the problem. Since and , by AA similarity. Similarly, .
Solution 2 (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)
Video Solution
Video Solution: https://www.youtube.com/watch?v=0AXF-5SsLc8
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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.