The apothem of a regular polygon is defined as the distance from the center of the polygon to any of its sides. In the case of an equilateral triangle, the apothem is the distance from the center to one of the sides.

To determine if the apothem of an equilateral triangle is half the height, let’s first define what the height of the triangle is. The height of an equilateral triangle is the distance from one of the vertices to the opposite side, measured along a perpendicular line.

Now, let’s consider an equilateral triangle with a side length of “s”. To find the height, we can divide the triangle into two congruent right triangles by drawing a perpendicular line from one of the vertices to the midpoint of the opposite side. This line is also known as the altitude of the triangle.

Using the Pythagorean theorem, we can calculate the height of the triangle. Let’s call the height “h”. In one of the right triangles, the hypotenuse is the side length “s”, and one of the legs is the height “h”. The other leg is half the side length, which is “s/2”. Applying the Pythagorean theorem, we have:

(s/2)^2 + h^2 = s^2

Simplifying this equation, we get:

S^2/4 + h^2 = s^2

Multiplying both sides by 4 to eliminate the denominator, we have:

S^2 + 4h^2 = 4s^2

Rearranging the equation, we get:

4h^2 = 3s^2

Dividing both sides by 4, we have:

H^2 = 3s^2/4

Taking the square root of both sides, we get:

H = √(3s^2/4)

So, the height of the equilateral triangle is given by the formula:

H = √(3s^2/4)

Now, let’s consider the apothem of the equilateral triangle. As mentioned earlier, the apothem is the distance from the center of the triangle to one of its sides. In an equilateral triangle, the apothem is also equal to the length of the median, which is the line segment connecting the center of the triangle to the midpoint of one of its sides.

To determine if the apothem is half the height, we need to compare the apothem to the height. Let’s call the apothem “a”. From the definition of the apothem, we know that it is perpendicular to one of the sides. Therefore, we can draw a right triangle with the apothem as one of the legs and the height as the other leg.

Using the Pythagorean theorem, we can calculate the length of the apothem. In the right triangle, the hypotenuse is the height “h”, one of the legs is the apothem “a”, and the other leg is half the side length “s/2”. Applying the Pythagorean theorem, we have:

(a)^2 + (s/2)^2 = h^2

Substituting the value of “h” from the previous equation, we have:

(a)^2 + (s/2)^2 = 3s^2/4

Multiplying through by 4 to eliminate the denominator, we have:

4(a)^2 + s^2 = 3s^2

Rearranging the equation, we get:

4(a)^2 = 2s^2

Dividing both sides by 4, we have:

(a)^2 = s^2/2

Taking the square root of both sides, we get:

A = √(s^2/2)

So, the length of the apothem of the equilateral triangle is given by the formula:

A = √(s^2/2)

Comparing the formulas for the height and the apothem, we have:

H = √(3s^2/4)

A = √(s^2/2)

From these formulas, we can see that the apothem is not half the height of the equilateral triangle. The ratio between the apothem and the height is:

A/h = √(s^2/2) / √(3s^2/4) = √(2/3)

Therefore, the apothem is equal to √(2/3) times the height of the equilateral triangle, not half the height.