To find the length of one side of an equilateral triangle given its altitude, we can use specific relationships that exist within an equilateral triangle.
1. Understanding the altitude of an equilateral triangle:
- The altitude of an equilateral triangle creates two 30-60-90 right triangles. In a 30-60-90 triangle, the relationships between the sides are well defined. If the length of the side opposite the 30° angle is [tex]\( x \)[/tex], the length of the side opposite the 60° angle (which is the altitude) will be [tex]\( x \sqrt{3} \)[/tex], and the hypotenuse (which is the side of the equilateral triangle) will be [tex]\( 2x \)[/tex].
2. Given:
- The altitude [tex]\( h \)[/tex] of the equilateral triangle is [tex]\( 7 \sqrt{3} \)[/tex].
3. Finding the side length:
- Using the relationship mentioned above, the altitude [tex]\( h = \frac{\sqrt{3}}{2} \times a \)[/tex], where [tex]\( a \)[/tex] is the side length of the triangle.
4. Isolate [tex]\( a \)[/tex]:
- Rearranging the formula to solve for [tex]\( a \)[/tex], we get [tex]\( a = \frac{2h}{ \sqrt{3}} \)[/tex].
- Substitute [tex]\( h = 7 \sqrt{3} \)[/tex] into the formula:
[tex]\[
a = \frac{2 \times 7 \sqrt{3}}{ \sqrt{3}}
\][/tex]
- Simplify the expression:
[tex]\[
a = \frac{14 \sqrt{3}}{ \sqrt{3}}
\][/tex]
[tex]\[
a = 14
\][/tex]
Therefore, the length of one side of the equilateral triangle is [tex]\( 14 \)[/tex] units. So, the answer is:
[tex]\[ 14 \][/tex]
