Difference between revisions of "2024 AMC 12A Problems/Problem 19"
(Created page with "==solution 1== <math>\angle CBA=60 ^\circ</math> by Circle Theorem} Let <math>AC=u</math>, apply cosine law on <math>\triangle ACD</math> <cmath>u^2=3^2+5^2-2(3)(5)cos120</cma...") |
(→solution 1) |
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| Line 13: | Line 13: | ||
<cmath>BD=\frac{39}{7}</cmath> | <cmath>BD=\frac{39}{7}</cmath> | ||
Since <math>\frac{39}{7}<5</math> | Since <math>\frac{39}{7}<5</math> | ||
| − | The answer is <math>\fbox{(D) | + | The answer is <math>\fbox{(D) \frac{39}{7}}</math> |
Revision as of 18:02, 8 November 2024
solution 1
by Circle Theorem}
Let 
, apply cosine law on 
![\[u^2=3^2+5^2-2(3)(5)cos120\]](http://latex.artofproblemsolving.com/c/b/9/cb9edf966b93fb32e04934643d7e38910d16d429.png)
![\[u=7\]](http://latex.artofproblemsolving.com/e/c/7/ec7657c1ae5d5e39fe5611d3dbf210b65d39c41f.png)
Let 
, apply cosine law on 
![\[7^2=3^2+v^2-2(3)(v)cos60\]](http://latex.artofproblemsolving.com/6/e/c/6ec08e130c69354c1c6f6bbb0c55eba73426234f.png)
![\[v=\frac{3\pm13}{2}\]](http://latex.artofproblemsolving.com/e/f/d/efd53a5925c1959130cfb0ccb9b4c0545adfd1a6.png)
![\[v=8\]](http://latex.artofproblemsolving.com/5/c/8/5c821fed96435f71d8b2086f2ac5d7a733229173.png)
By Ptolemy Theorem,
![\[AB \cdot CD+AD \cdot BC=AC \cdot BD\]](http://latex.artofproblemsolving.com/8/f/9/8f90407491cc1fcb3e335391dabab417990ba5c0.png)
![\[8 \cdot 3+5 \cdot 3=7BD\]](http://latex.artofproblemsolving.com/9/4/9/9498c005ba103e2bb5e799ae3d3a05f0e615b88a.png)
![\[BD=\frac{39}{7}\]](http://latex.artofproblemsolving.com/c/5/1/c516c5649962b98d6b4f2fa26c9ebeba2eefde3a.png)
Since 
The answer is $\fbox{(D) \frac{39}{7}}$ (Error compiling LaTeX. Unknown error_msg)
