If an equilateral triangle has a side of 10 ft what is its altitude to two significant digits
To find the altitude of an equilateral triangle, you can use the Pythagorean theorem or trigonometry.
1. Using the Pythagorean theorem:
- An equilateral triangle is a special type of triangle with all sides and angles equal.
- The altitude of an equilateral triangle cuts the triangle into two right triangles.
- To find the altitude, we can divide the triangle into two right triangles by drawing a perpendicular line from one vertex to the opposite side.
- The altitude acts as the hypotenuse of one of the right triangles.
- The base of the right triangle is half the length of one side of the equilateral triangle (since the equilateral triangle has all sides equal).
- Using the Pythagorean theorem, we can find the length of the altitude.
Let's calculate:
- One side of the equilateral triangle is 10 ft.
- The base of the right triangle is half the length of one side, which is 10 ft / 2 = 5 ft.
- Using the Pythagorean theorem, we have: altitude^2 = (side^2) - (base^2)
- altitude^2 = 10^2 - 5^2 = 100 - 25 = 75
- altitude = √75 ≈ 8.66 ft (rounded to two significant digits)
2. Using Trigonometry:
- Another way to find the altitude of an equilateral triangle is by using trigonometry.
- Since all sides and angles of an equilateral triangle are equal, each angle of the triangle is 60 degrees.
- By drawing an altitude, we form a right triangle with one angle measuring 60 degrees.
- Using trigonometry, we can calculate the length of the altitude.
Let's calculate:
- In the right triangle, the opposite side is the altitude, and the adjacent side is half the length of one side of the equilateral triangle (as calculated before, 5 ft in this case).
- Using the trigonometric function for the tangent of an angle, we have: tan(60 degrees) = opposite / adjacent
- tan(60 degrees) = altitude / 5 ft
- altitude = 5 ft * tan(60 degrees) ≈ 8.66 ft (rounded to two significant digits)
Therefore, the altitude of the equilateral triangle with a side length of 10 ft is approximately 8.66 ft (rounded to two significant digits).