# The height of an equilateral triangle is 12 √3. Find a side of it?

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

Thus, the height can be drawn to either side.

The sum of all the angles of any triangle is 180 °. This means that the magnitude of each angle of an equilateral triangle is 60 °. (180 ÷ 3 = 60 °).

Let’s draw an equilateral triangle ABC. If we draw the height of CH in it (this is the perpendicular lowered to the side AB), we get two right-angled triangles ACН and BCH.

Find the sine of angle A (angle A is 60 °).

From the definition of sine: sine is the ratio of the opposite leg to the hypotenuse. Let’s write down:

Sin 60 ° = h / a,

where a is the side of the triangle, h is the height drawn to this side.

Let us express the side of the triangle from the formula, and:

a = h / Sin 60 °.

From the trigonometric table: Sin 60 ° = √3 / 2, substitute the sine value in the resulting expression for the side of the triangle:

a = h / (√3 / 2) = 2 h / √3.

Let’s calculate the side of the triangle:

a = 2 h / √3 = (2 × 12√3) / √3 = (24 √3) / √3 = 24.

Answer: the side of an equilateral triangle ABC is 24.