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# Helpp

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A regular hexagon ABCDEF has area 36 units squared. What is the area of triangle ACE? Thanks in advance
Jun 12, 2022

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We know that each angle in the hexagon is $$120 ^ \circ$$

By symmetry, we know that $$\overline {AC} = \overline{CE} = \overline { AE}$$, meaning $$\triangle ACE$$(red in diagram) is equilateral.

Now, let the center of the triangle be G, and draw lines (blue) connecting it to Points A, E, and C.

This divides it into 3 isosceles triangles, each with 1 $${120 ^ \circ}$$ and 2 $${30 ^ \circ}$$ angles.

Note that these triangles are congruent to $$\triangle AFE, \triangle CDE,$$ and $$\triangle ABC$$

This means that $$\triangle ACE$$ contains 3 out of the 6 triangles, meaning its area is $$0.5 \times 36 = \color{brown}\boxed{18}$$

Jun 13, 2022