Question

Find a side of an equilateral triangle with a 12 inch altitude

Answer 1

To find the side of an equilateral triangle with a 12 inch altitude, we use the 1:√3:2 ratio of sides in a 30-60-90 triangle, which results in a side length of 8√3 inches.

Step-by-step explanation:

To find the side of an equilateral triangle with a 12 inch altitude, we will apply the Pythagorean theorem. In an equilateral triangle, if you draw an altitude, it will split the triangle into two 30-60-90 right triangles. In such a triangle, the altitude represents the longer leg, the half of the side of the equilateral triangle represents the shorter leg, and the full side of the triangle represents the hypotenuse.

For a 30-60-90 right triangle, the lengths of the sides are in the ratio 1:√3:2. Therefore, if the altitude (longer leg) is 12 inches, then the half of the side of the triangle (shorter leg) is 12/√3 inches. To find the full length of the side (hypotenuse), we multiply the shorter leg by 2, which gives us the formula:

Side = 2 * (12/√3) = 24/√3 = 8√3 inches

Thus, the side of the equilateral triangle is 8√3 inches.

Answer 2

Since the triangle is an equilateral triangle we know all of it's sides must be the same length, with that in mind the angles that make up the triangle must be equal as well. Knowing that a triangle's three interior angles make up 180 degrees we know that the size of each angle must be one third of this (as each angle must be equal).

180/3 = 60


then we may split the triangle along it's altitude into two special right triangles
more specifically two 30-60-90 triangles.

this means that the side with 30 degrees will be some value "x" where the side for 60 degrees will be related as it is "x*sqrt(3)" and the hypotenuse (which would be the side of the triangle) would be proportionally "2x"

this would mean that the altitude is the side associated with the 60 degree angle as such we can solve for "x" using this.

12= x*sqrt(3)
12/sqrt(3)=x
4sqrt(3)=x (simplifying the radical we get "x" equals 4 square root 3)

now we may solve for the side length of the triangle which is "2x"

2*4sqrt(3) -> 8sqrt (3)

eight square root of three is the answer.