An equilateral triangle with altitude length of 7 square root 3 centimeters.
Find the area and perimeter
A=49 square root 3 divided by 2 cm ^2
P=42cm
h = 7√3, so each side is 14
A = (1/2)(14)(7√3) = 49√3
P = 3*14 = 42
To find the area of an equilateral triangle, you can use the formula A = (√3/4) * a^2, where "a" is the length of one side of the triangle.
Given that the altitude length of the equilateral triangle is 7√3 cm, we know that the altitude splits the equilateral triangle into two congruent right triangles. The hypotenuse of each right triangle is one side of the equilateral triangle, and the altitude is the height of the right triangle. By using the Pythagorean theorem, we can find the length of one side of the triangle.
Let's denote the side length of the equilateral triangle as "x". We have the formula x^2 = (7√3)^2 - (x/2)^2.
Simplifying, we get:
x^2 = 147 - (x^2/4)
4x^2 = 588 - x^2
5x^2 = 588
x^2 = 588/5
x = √(588/5)
x = (2√147)/√5
x = 2√(147/5)
x = 2√(29.4)
x ≈ 13.62 cm
Now that we know the length of one side of the equilateral triangle is approximately 13.62 cm, we can calculate the area and perimeter.
Area (A) = (√3/4) * a^2
A = (√3/4) * (13.62)^2
A ≈ (√3/4) * 185.33
A ≈ 49√3 cm^2
Perimeter (P) of an equilateral triangle is P = 3 * a, where "a" is the length of one side.
P = 3 * 13.62
P ≈ 40.86 cm
Therefore, the area of the equilateral triangle is approximately 49√3 cm^2 and the perimeter is approximately 40.86 cm.