Geometry

Find AE.

$\triangle DEF\ is\ equilateral. \\ DA = 3,\ AF = 2,\ EF=5\\ \angle AFE=60°$

Pythagorean trigonometric theorem: $a²=b²+c²-2bc\cdot cos\ \alpha$

$\overline{AE}\ ^2=\overline{AF}\ ^2+\overline{EF}\ ^2-cos\ (AFE)\\\overline{AE}\ ^2=2^2+5^2-2\cdot 5\cdot cos\ 60^°\\ \overline{AE}= \sqrt{29-2\cdot 5\cdot0.5}\\ \color{blue }\overline{AE}= \sqrt{24}=4.899$

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