an equilateral triangle has an altitude length of 36 feet. determine the length of a side of the triangle
The altitude will be the opposite side to 60° angle of a 30-60-90 right-angled triangle, where the hypotenuse is the length of the equilateral triangle
let that side be x
sin 60° = 36/x
x = 36/sin60 = 36/(√3/2) = 72/√3 = appr. 41.6 feet
To determine the length of a side of an equilateral triangle, we can use the altitude length. Here's how you can find the solution:
1. In an equilateral triangle, the altitude divides the triangle into two congruent right triangles.
2. Since the triangle is equilateral, all three angles are equal and each angle measures 60 degrees.
3. In a right triangle, the altitude is the height of the triangle and forms a right angle with the base.
4. In an equilateral triangle, the altitude bisects the base into two congruent segments.
5. The altitude of an equilateral triangle is also a median and an angle bisector.
6. Using the properties of a right triangle, we can form a right triangle by drawing a line segment from one vertex to the midpoint of the base, creating a right angle with the base.
7. Let's call the length of the base of the equilateral triangle "x." Since the altitude bisects the base, each half of the base would be x/2.
8. Now, we have a right triangle with a base of x/2, a height of 36 feet, and a hypotenuse equal to a side length of the equilateral triangle (which we want to find).
9. Using the Pythagorean theorem, we can solve for the unknown side length. The Pythagorean theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.
In this case, (x/2)^2 + 36^2 = x^2.
10. Simplifying the equation, we get x^2/4 + 1296 = x^2.
11. To eliminate the fractions, we can multiply both sides by 4 to get x^2 + 5184 = 4x^2.
12. Rearranging the equation, we have 3x^2 = 5184.
13. Divide both sides by 3 to solve for x^2: x^2 = 1728.
14. Taking the square root of both sides, we find x = √1728.
15. Simplifying, we get x ≈ 41.57 feet.
Therefore, the length of a side of the equilateral triangle is approximately 41.57 feet.
To determine the length of a side of an equilateral triangle, we need to use the formula for the length of the altitude.
Let's assume "s" represents the length of a side of the triangle.
In an equilateral triangle, the altitude bisects the base into two equal segments, forming two congruent right triangles. The altitude also serves as the hypotenuse of one of those right triangles.
Using the Pythagorean theorem, we can find the length of the base (which is half of a side of the triangle):
b^2 + h^2 = s^2
Since the altitude bisects the base, half of the base is 36/2 = 18 feet.
Therefore, the equation becomes:
18^2 + 36^2 = s^2
324 + 1296 = s^2
1620 = s^2
To find the length of a side, we take the square root of both sides:
√1620 = √(s^2)
So, s ≈ 40.25 feet.
Therefore, the length of a side of the equilateral triangle is approximately 40.25 feet.