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2024 AMC 12B Problems/Problem 22: Difference between revisions

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==Solution 1==
==Solution 1==


We will use typical naming for the sides and angles of the triangle, that is <math>AB=</math>
Let <math>AB=c</math>, <math>BC=a</math>, <math>AC=b</math>. According to the law of sines,
<cmath>\frac{b}{a}=\frac{\sin \angle B}{\sin \angle A}</cmath>
<cmath>=2\cos</cmath>

Revision as of 03:32, 14 November 2024

Problem 22

Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle?

$\textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad$

Solution 1

Let $AB=c$, $BC=a$, $AC=b$. According to the law of sines, \[\frac{b}{a}=\frac{\sin \angle B}{\sin \angle A}\] \[=2\cos\]

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