The median of an equilateral triangle is 11√3. Find a side of it.

Based on the property of an equilateral triangle, we can say that in an equilateral triangle all sides and angles are equal to each other.

Property of the median of an equilateral triangle: “In an equilateral triangle, the median drawn to either side is also its bisector and height.”

Let’s designate the side of our equilateral triangle by the letter a.

This means that the median, or height, divides our equilateral triangle into two rectangular ones. Let’s consider one of them.

The hypotenuse of this right-angled triangle is side a, one leg is 11√3, and the other is a / 2 (property of the median).

Using the Pythagorean theorem, we write down the expression for a given right-angled triangle and find the side a from it. Pythagorean theorem: “The square of the hypotenuse is equal to the sum of the squares of the legs.”

c² + b² = a²,

where c and b are the legs of a right-angled triangle, a is the hypotenuse.

(11√3) ² + (a / 2) ² = a²;

11² × 3 + a² / 4 = a²;

11² × 3 = a² – a² / 4;

11² × 3 = (4а² – а²) / 4;

11² × 3 = 3а² / 4;

(4 × (11² × 3)) / 3 = a²;

a = √ (4 × 11²);

a = √ (2² × 11²) = 22.

Let’s check:

22² = (11√3) ² + (22/2) ² = 121 × 3 + 121 = 484.

484 = 484.

We decided correctly.

Answer: the side of an equilateral triangle is 22.