What is the length of an equilateral triangle whose altitude has a length of 21

Let each side be 2x

then we would have a right-angled triangle with sides, x, 21, and 2x so that
x^2 + 21^2 = (2x)^2
3x^2= 441
x^2 = 147
x = √147 = 7√3

so each side of the equilateral triangle is 14√3

or using trig

let each side be s
then 21/s =sin60°
s = 21/sin60° = 21/(√/2) = 42/√3
= 14√3 after rationalizing the denominator, same asnwer as above.

To find the length of an equilateral triangle, we can use the relationship between the altitude and the side length.

In an equilateral triangle, the altitude drawn from any vertex bisects the base into two congruent segments, forming a right triangle with the base.

Let's call the length of the altitude "h" and the side length "s". In this case, h = 21.

Using the Pythagorean theorem, we can find the length of the base (s), which is also the side length of the equilateral triangle.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

In this case, the base is one of the legs of the right triangle, and the altitude is the other leg.

So, we have:
s^2 = h^2 + (0.5s)^2

where 0.5s is half the side length since it represents half the base when the altitude bisects it.

Plugging in the values, we get:
s^2 = 21^2 + (0.5s)^2

Expanding, we have:
s^2 = 441 + 0.25s^2

Combining like terms, we have:
0.75s^2 = 441

Dividing both sides by 0.75, we get:
s^2 = 588

Taking the square root of both sides, we get:
s ≈ √588

Approximately, s ≈ 24.25

Therefore, the length of the equilateral triangle is approximately 24.25 units.

To find the length of an equilateral triangle given its altitude, we can use the formula:

length = 2 * (altitude / √3)

In this case, the altitude of the equilateral triangle is given as 21. So, the length would be:

length = 2 * (21 / √3)
= 2 * (21 / √3) * (√3 / √3)
= 2 * (21√3 / 3)
= 42√3 / 3
≈ 24.25

Therefore, the length of the equilateral triangle is approximately 24.25.