the perimeter of an equilateral triangle is 32 centimeters. find the length of an altitude of the triangleto the nearest tenth of a centimeter.
A=10.6cm
H^2=84.27
H=9.1 cm
To find the length of the altitude of an equilateral triangle, we need to use the property that the altitude of an equilateral triangle forms a right triangle with one of the sides of the triangle.
Let's denote the length of one side of the equilateral triangle as 's' and the length of the altitude as 'h'.
Since an equilateral triangle has three equal sides, each side of the triangle is the same length.
Therefore, the perimeter of the triangle is given by:
Perimeter = 3s
We are given that the perimeter of the triangle is 32 centimeters.
So, we have the equation:
3s = 32
To find the length of s, we can divide both sides of the equation by 3:
s = 32 / 3
Now, since the altitude forms a right triangle with one of the sides, we can use the Pythagorean theorem to find the length of the altitude.
The Pythagorean theorem states:
h^2 = s^2 - (s/2)^2
Substituting the value of s, we have:
h^2 = (32/3)^2 - ((32/3)/2)^2
Calculating this, we get:
h^2 = 1024/9 - (64/9)
h^2 = 960/9
h^2 = 106.67
To find the length of h, we take the square root of both sides:
h ≈ √106.67
h ≈ 10.33 centimeters (rounded to the nearest tenth of a centimeter)
Therefore, the length of the altitude of the equilateral triangle is approximately 10.33 centimeters.