Triangle DEF is an equilateral triangle with a side length of 12 cm. If point G is the mid-point of side DE, what is the length (in cm) of side FG?

The correct option is:

$6\sqrt{3}$

Step-by-step Explanation:

1. Understand the properties of the triangle:
We are given that triangle DEF is an equilateral triangle with a side length of 12 cm. In an equilateral triangle, all three sides are equal in length. Therefore, we have:

$\mathrm{DE}=\mathrm{EF}=\mathrm{FD}=12\text{cm}$

2. Determine the length of the split segment:
Point G is the midpoint of side DE. Since G divides DE into two equal halves, the length of segment DG (and GE) is:

$\mathrm{DG}=\frac{\mathrm{DE}}{2}=\frac{12}{2}=6\text{cm}$

3. Analyze the geometry:
In any equilateral triangle, the line segment connecting a vertex to the midpoint of the opposite side (the median) also acts as the altitude, meaning it is perpendicular to that side. Thus, the segment FG is perpendicular to DE, creating a right-angled triangle FGD (with the right angle at G).

4. Apply the Pythagorean Theorem:
In the right-angled triangle FGD, the hypotenuse is FD, and the other two sides are DG and FG. According to the Pythagorean theorem:

${\mathrm{FD}}^2={\mathrm{DG}}^2+{\mathrm{FG}}^2$

Substituting the known lengths into the equation:

${12}^2=6^2+{\mathrm{FG}}^2$

$144=36+{\mathrm{FG}}^2$

Subtract 36 from both sides to isolate FG²:

${\mathrm{FG}}^2=144-36$

${\mathrm{FG}}^2=108$

Take the square root of both sides to find FG:

$\mathrm{FG}=\sqrt{108}$

Simplify the radical by factoring out the largest perfect square (36):

$\mathrm{FG}=\sqrt{36\times 3}=6\sqrt{3}\text{cm}$

Consequently, the length of side FG is 6√3 cm.