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Triangle perimeter

Source: 2019 AMC 12A #19

February 8, 2019
trigonometrygeometryperimeter2019 AMC 12AAMC2019 AMCAMC 12

Problem Statement

In △ABC\triangle ABC with integer side lengths, cos⁡A=1116,cos⁡B=78,andcos⁡C=−14. \cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}. What is the least possible perimeter for △ABC\triangle ABC?
<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>12<spanclass=′latex−bold′>(C)</span>23<spanclass=′latex−bold′>(D)</span>27<spanclass=′latex−bold′>(E)</span>44<span class='latex-bold'>(A) </span> 9 \qquad <span class='latex-bold'>(B) </span> 12 \qquad <span class='latex-bold'>(C) </span> 23 \qquad <span class='latex-bold'>(D) </span> 27 \qquad <span class='latex-bold'>(E) </span> 44
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