Why is the apothem of an equilateral triangle 1/3 its height?
To understand why the apothem of an equilateral triangle is one-third of its height, we need to break down the components of an equilateral triangle and use geometric properties.
To solve this, we can draw an equilateral triangle and label its sides, height, and apothem. Let's assume that the height of the triangle is 'h,' and the apothem is 'a'.
Now, we can divide the equilateral triangle into two congruent right triangles by drawing a line from the apex (top vertex) to the midpoint of the base. This line segment will be perpendicular to the base and intersect it at its midpoint, forming right angles.
The height of the equilateral triangle will also be the height of each right triangle formed. Thus, each right triangle will have a height of 'h'.
Next, let's focus on one of the right triangles formed in the equilateral triangle. The base of the right triangle will be half of the base of the equilateral triangle, which is equal to 'h/2.' The apothem of the equilateral triangle will also be the hypotenuse of the right triangle, which is equal to 'a'.
We can now apply the Pythagorean theorem to the right triangle to relate its sides. The theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using the Pythagorean theorem, we have:
(a)^2 = (h/2)^2 + (h)^2
Simplifying it further:
a^2 = h^2/4 + h^2
a^2 = (h^2 + 4h^2)/4
a^2 = 5h^2/4
To find the relationship between the apothem and the height, we can find the square root of both sides:
√(a^2) = √(5h^2/4)
a = h√5/2
Now, we have the apothem 'a' in terms of 'h'. To prove that the apothem is one-third of the height, we can compare the two values:
a/h = h√5/2 / h
a/h = (√5/2)
As we can see, the ratio of the apothem (a) to the height (h) is (√5/2). To simplify this ratio:
(√5/2) ≈ 0.79
Therefore, the apothem of an equilateral triangle is approximately 0.79 times its height. Rounded to the nearest whole number, it is approximately one-third. Hence, we can conclude that the apothem of an equilateral triangle is about one-third of its height.