2024 AMC 12A Problems/Problem 19

solution 1

$\angle CBA=60 ^\circ$ by Circle Theorem} Let $AC=u$ , apply cosine law on $\triangle ACD$ $u^2=3^2+5^2-2(3)(5)cos120$ $u=7$ Let $AB=v$ , apply cosine law on $\triangle ABC$ $7^2=3^2+v^2-2(3)(v)cos60$ $v=\frac{3\pm13}{2}$ $v=8$ By Ptolemy Theorem, $AB \cdot CD+AD \cdot BC=AC \cdot BD$ $8 \cdot 3+5 \cdot 3=7BD$ $BD=\frac{39}{7}$ Since $\frac{39}{7}<5$ The answer is $\fbox{(D) \frac{39}{7}}$ (Error compiling LaTeX. Unknown error_msg)

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2024 AMC 12A Problems/Problem 19

solution 1