An equilateral triangle is a geometric shape that has three equal sides and three equal angles, each measuring 60 degrees. It is a special case of an isosceles triangle, where two sides are equal, and an acute triangle, where all angles are less than 90 degrees.
The altitude, median, and angle bisectors of an equilateral triangle are the same line.
The circumcenter and incenter of an equilateral triangle coincide, and the distance from each vertex to the circumcenter is equal to the length of the side of the triangle.
The area of an equilateral triangle can be found using the formula A = (sqrt(3)/4)*s^2, where A is the area and s is the length of one side.
Now, let’s look at some examples of problems involving equilateral triangles:
Example 1: Find the perimeter of an equilateral triangle with a side length of 6 cm.
Solution: Since all sides are equal, the perimeter of the triangle is simply 3 times the length of one side. Therefore, the perimeter is:
P = 3s = 3(6) = 18 cm
Therefore, the perimeter of the equilateral triangle is 18 cm.
Example 2: Find the area of an equilateral triangle with a side length of 8 cm.
Solution: The area of an equilateral triangle can be found using the formula A = (sqrt(3)/4)*s^2, where A is the area and s is the length of one side. Therefore, substituting the values given in the problem, we get:
A = (sqrt(3)/4)*8^2
A = (sqrt(3)/4)*64
A = 16sqrt(3)
Therefore, the area of the equilateral triangle is 16sqrt(3) square centimeters.
Example 3: Find the length of the altitude of an equilateral triangle with a side length of 12 cm.
Solution: In an equilateral triangle, the altitude is also the median and the angle bisector. Therefore, we can divide the triangle into two 30-60-90 triangles, where the altitude is the longer leg and the hypotenuse is the side of the equilateral triangle. In a 30-60-90 triangle, the longer leg is equal to (sqrt(3)/2)*the hypotenuse. Therefore, we can find the length of the altitude by:
Altitude = (sqrt(3)/2)*s
Altitude = (sqrt(3)/2)*12
Altitude = 6sqrt(3)
Therefore, the length of the altitude of the equilateral triangle is 6sqrt(3) cm.
Example 4: Find the length of one side of an equilateral triangle with an area of 48 square units.
Solution: We know that the area of an equilateral triangle can be found using the formula A = (sqrt(3)/4)*s^2. Therefore, we can rearrange the formula to solve for s:
s = sqrt(4A/sqrt(3))
Substituting the given value of the area, we get:
s = sqrt(4(48)/sqrt(3))
s = 8sqrt(3)
Therefore, the length of one side of the equilateral triangle is 8sqrt(3) units.
These examples demonstrate how to solve various problems involving equilateral triangles using their properties and formulas.
