Final answer:
The perimeter of the equilateral triangle with a height of 12 cm is 24√3 cm, and the area is 48√3 cm² after using the properties of a 30-60-90 right triangle to determine the side length.
Step-by-step explanation:
To identify the perimeter and area of an equilateral triangle with a height of 12 cm, we first need to find the length of one side. Since the triangle is equilateral, all sides are equal. To find the side length, we can use the fact that in an equilateral triangle, the height forms a 30-60-90 right triangle with half of one side and the entire height.
Let s be the length of one side of the equilateral triangle. The height (12 cm) corresponds to the long leg of the 30-60-90 triangle, which is s√3/2. We can set up an equation:
12 = s√3/2 and solve for s. Multiplying both sides by 2/√3, we get s = 24/√3 = 8√3 cm.
Now we know the side length of the triangle is 8√3 cm. To find the perimeter, we multiply this side length by 3 since there are three equal sides in the triangle. The perimeter P = 3 * 8√3 = 24√3 cm.
To find the area A, we use the formula 1/2 × base × height. Since the base is one side of the triangle, it's 8√3 cm, and the height is given as 12 cm. Therefore, the area A = 1/2 × 8√3 × 12 = 48√3 cm².