To find the length of the altitude of an equilateral triangle with sides of 8 units, we start with the properties of an equilateral triangle.
1. Properties of Equilateral Triangle:
- All three sides are equal in length.
- All three interior angles are 60 degrees.
- The altitude splits the equilateral triangle into two 30-60-90 right triangles.
2. Altitude of an Equilateral Triangle:
- In a 30-60-90 triangle, the altitude (which we are solving for) corresponds to the longer leg.
- We recall that the ratios of the sides in a 30-60-90 triangle are [tex]\(1 : \sqrt{3} : 2\)[/tex], where:
- The length of the shorter leg (opposite 30 degrees) is half the length of the side of the equilateral triangle.
- The length of the longer leg (opposite 60 degrees) is the shorter leg times [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is the same as the side of the equilateral triangle.
3. Calculation Steps:
- For our triangle, the side length (hypotenuse) of the 30-60-90 triangle is 8 units.
- The shorter leg is [tex]\( \frac{8}{2} = 4 \)[/tex] units.
- The longer leg (altitude) is [tex]\( 4 \times \sqrt{3} = 4\sqrt{3} \)[/tex] units.
Given the choices:
- [tex]\(5 \sqrt{2}\)[/tex] units
- [tex]\(4 \sqrt{3}\)[/tex] units
- [tex]\(10 \sqrt{2}\)[/tex] units
- [tex]\(16 \sqrt{5}\)[/tex] units
The correct length of the altitude is [tex]\(4 \sqrt{3}\)[/tex] units.
So, the length of the altitude is [tex]\(4 \sqrt{3}\)[/tex] units.
