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# 2 probs. Help quick

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A square tile, a regular hexagon tile, and a regular \$n\$-sided polygon tile share a common vertex. There are no gaps or overlaps between the tiles. What is \$n?\$

The exterior angles of a \$k\$-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are \$136^{\circ}\$ and \$176^{\circ}\$. What is \$k?\$

Feb 5, 2021

### 1+0 Answers

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1.  The sum of the angles surrounding the common vertex is \$360^\circ\$. The right angle from the square corner is \$90^\circ\$ of that, and the interior angle of the hexagon is another \$120^\circ\$. Thus, the remaining angle is \$360^\circ-90^\circ-120^\circ=150^\circ\$. Since \$150^{\circ}\$ is an interior angle of the \$n\$-gon, each exterior angle measures \$30^{\circ}\$. Thus, the number of vertices is \$\dfrac{360^{\circ}}{30^{\circ}}=\boxed{12}\$.

2.  The largest and smallest exterior angles measure \$44^{\circ}\$ and \$4^{\circ}\$. Since the exterior angles are in an arithmetic progression, the average of all of them is equal to the average of the largest and smallest. Since we have \$k\$ angles in total, and their sum is \$360^{\circ},\$ this means \$\$\dfrac{360^{\circ}}{k}=\dfrac{44^{\circ}+4^{\circ}}{2}=24^{\circ},\$\$so \$k=\dfrac{360^{\circ}}{24^{\circ}}=\boxed{15}\$.

Feb 5, 2021