Question

Навколо рівностороннього трикутника описано коло радіус 4 см. Знайдіть площу цього трикутника

Answer 1

О - центр опис навколо трикутника кола - точка перетину медіан, в прав трик. вони будуть і висотами. Нехай ВН - одна з медіан( висот) трикутника. Тоді ВО = 4 см. За вл. медіан: ВО:ОН=2:1, отже 4:ОН=2:1; ОН = 2.
ВН= ВО+ОН=6.
Розглянемо трик ВНС( кут Н=90°):
tg кута ВСН(=60° як кут прав трик) = ВН:СН, звідси CH= ВН: tg60°= 6: корень(3)= 2коренів(3).
За теор. Піфагора:
ВС= корень( 36+ 12)= 4корня(3)

S трик АВС = 1/2ВН*ВС = 12коренів(3)

Answer 2

Calculation of the Area of an Equilateral Triangle Inscribed in a Circle

To find the area of an equilateral triangle inscribed in a circle, we can use the formula:

Area of an equilateral triangle = (side length)^2 * sqrt(3) / 4

In this case, we are given that the radius of the circle is 4 cm. Since the triangle is inscribed in the circle, the radius of the circle is also the distance from the center of the circle to any of the vertices of the triangle.

Let's denote the side length of the equilateral triangle as 's'. We can find the side length using the formula:

Side length = 2 * (radius) * sin(60°)

Since the radius is given as 4 cm, we can substitute this value into the formula:

Side length = 2 * 4 cm * sin(60°)

Using the trigonometric identity sin(60°) = sqrt(3) / 2, we can simplify the equation:

Side length = 2 * 4 cm * sqrt(3) / 2

Simplifying further:

Side length = 4 cm * sqrt(3)

Now that we have the side length, we can calculate the area of the equilateral triangle using the formula mentioned earlier:

Area of the equilateral triangle = (4 cm * sqrt(3))^2 * sqrt(3) / 4

Simplifying further:

Area of the equilateral triangle = 16 cm^2 * 3^(3/2) / 4

Simplifying the expression:

Area of the equilateral triangle = 12 cm^2 * sqrt(3)

Therefore, the area of the equilateral triangle inscribed in a circle with a radius of 4 cm is 12 cm^2 * sqrt(3).

Please note that the calculations provided above are based on mathematical formulas and assumptions.