1. ## geometry help

An equilateral triangle is inscribed in a regular hexagon . This triangle's vertices touch the midpoints of three sides of a regular hexagon . The area of the triangle is $12cm^2$ . What is the area of the regular hexagon in $cm^2$ ?

I made a sketch . . . and the answer jumped out at me!

An equilateral triangle is inscribed in a regular hexagon.
This triangle's vertices touch the midpoints of three sides of the regular hexagon.
The area of the triangle is 12 cm².
What is the area of the regular hexagon in cm² ?

The diagram looks like this:
Code:
                *---*---*
/   / \   \
/   /   \   \
/   /     \   \
*   /       \   *
\ /         \ /
*-----------*
\         /
*-------*

Divide the hexagon into "unit triangles",
. . and we have this diagram:
Code:
                *---*---*
/ \ /x\ / \
*---*---*---*
/ \ /x\x/x\ / \
*---*---*---*---*
\ /x\x/x\x/x\ /
*---*---*---*
\ / \ / \ /
*---*---*

We see that the triangle occupies $\frac{9}{24} = \frac{3}{8}$ of the hexagon.

Therefore, the hexagon has: . $\frac{8}{3}\times12 \:=\:32\text{ cm}^2$