To find the length of the altitude of an equilateral triangle with side length 8 units, we can use the geometric properties of equilateral triangles.
1. Understanding the properties of the equilateral triangle:
- In an equilateral triangle, all sides are equal, and all angles are 60 degrees.
- The altitude of an equilateral triangle splits the triangle into two 30-60-90 right triangles.
2. Relationship in a 30-60-90 triangle:
- In a 30-60-90 triangle, the ratios of the sides are:
- The side opposite the 30° angle is the shortest side.
- The side opposite the 60° angle (which is the altitude in our case) is equal to the shortest side times [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse (which in our case is the side of the equilateral triangle) is twice the shortest side.
3. Determine the lengths of sides in the right triangle:
- The hypotenuse is the side of the equilateral triangle, which is 8 units.
- The shortest side (half of the equilateral triangle's side) is [tex]\( \frac{8}{2} = 4 \)[/tex] units.
4. Calculate the altitude:
- Using the property of the 30-60-90 triangle, the altitude is the shortest side times [tex]\(\sqrt{3}\)[/tex].
- Therefore, the altitude is [tex]\( 4 \times \sqrt{3} \)[/tex].
Using the above reasoning, the altitude of this equilateral triangle is:
[tex]\[ 4 \sqrt{3} \text{ units} \][/tex]
So, the correct answer is [tex]\( 4 \sqrt{3} \)[/tex] units.
