Final answer:
Using the properties of a 30-60-90 triangle and the Pythagorean theorem, the length of each side of an equilateral triangle with a height of 5√7 cm is 10√7 cm. Therefore, the perimeter is 30√7 cm.
Step-by-step explanation:
To find the length of the sides of an equilateral triangle with a height of 5√7 cm, we use the Pythagorean theorem. An equilateral triangle can be divided into two 30-60-90 right triangles, where the height is the shorter leg (opposite the 60-degree angle), the half of one side is the longer leg (opposite the 30-degree angle), and the full length of the side is the hypotenuse.
Using the properties of a 30-60-90 triangle, we know that the hypotenuse is twice the shorter leg, so if the height (shorter leg) is 5√7 cm, the length of the side (hypotenuse) is double that:
Side length = 2 × height = 2 × 5√7 cm = 10√7 cm.
To find the perimeter, we just multiply the length of one side by 3, since an equilateral triangle has three equal sides:
Perimeter = Side length × 3 = 10√7 cm × 3 = 30√7 cm.