The height of an equilateral triangle is 9√3. Find a side of it.

We need to find the length of the side of an equilateral triangle, the median length of which is 9√3.

By the property of an equilateral triangle, the median divides the side to which it is drawn into two equal segments. And at the same time it is height.

As a result, we get two identical right-angled triangles.

In a right-angled triangle, the hypotenuse (side of an equilateral triangle) will be equal to x, one of the legs is equal to x / 2 (half of the side to which the median is drawn), the second leg is the median equal to 3 cm.

We apply the Pythagorean theorem and get the equation.

x ^ 2 = (x / 2) ^ 2 + (9√3) ^ 2;

x ^ 2 = 243 + x ^ 2/4;

4x ^ 2 = 972 + x ^ 2;

4x ^ 2 – x ^ 2 = 972;

3x ^ 2 = 972;

x ^ 2 = 972: 3;

x ^ 2 = 324;

x = √324;

x = 18 is the length of the side of the triangle.