Question

If the length of the median of an equilateral triangle is $3\sqrt{3}$ then the length of a side of the triangle of _______.

• A. $9$
• B. $6$
• C. $7.5$
• D. $3$

Answer 1

The correct option is B $6$

The length of the median of an equilateral triangle is $3\sqrt{3}$.
Let the measure of each side of an equilateral triangle is $x$.
$\therefore AB=BC=AC=x$
$\overline{AM}$ is a median. $AM=3\sqrt{3}$
$\therefore BM=CM=\frac{x}{2}$
In an equilateral triangle, median is an altitade.
$\therefore \overline{AM}\perp \overline{BC}$
$△AMB$ is a right angle triangle.
$\therefore A{B}^2=A{M}^2+B{M}^2$
$\therefore x^2=(3\sqrt{3}{)}^2+{\left(\frac{x}{2}\right)}^2$
$\therefore x^2=27+\frac{x^2}{4}$
$\therefore x^2-\frac{x^2}{4}=27$
$\therefore \frac{3x^2}{4}=27$
$\therefore x^2=27\times \frac{4}{3}=36$
$\therefore x=6$
Therefore, the length of side of the triangle is $6$.