The top of a valentine candy box is in the shape of an equilateral triangle if the altitude of the triangle is 8cm what is the perimeter of the triangle
let the side of the triangle be 2x
then tan60 = 8/x
x = 8/tan60
perimeter = 3(2x) = 6x
= 48/√3 = appr 27.7 cm
To find the perimeter of an equilateral triangle, we need to know the length of one side. In this case, we are given the altitude of the triangle, which is the length from one vertex to the opposite side.
Since the triangle is equilateral, all three sides have the same length. To find the length of one side, we can use the altitude as well as the fact that an equilateral triangle is composed of two congruent right triangles.
Let's use the Pythagorean theorem to find the length of one side using the altitude:
1. Draw one side of the equilateral triangle and the altitude from one of the vertices.
2. Label the length of the altitude as 8 cm. This altitude forms a right angle with the side of the triangle it intersects.
3. Divide the triangle into two right triangles by drawing a line segment from the vertex to the midpoint of the base. The length of this segment will be half the length of one side of the equilateral triangle.
4. Label the length of the segment connecting the vertex to the midpoint of the base as "x" cm.
5. Apply the Pythagorean theorem to one of the right triangles:
a^2 + b^2 = c^2
(8 cm)^2 + (x cm)^2 = (s cm)^2, where "s" represents the length of one side.
64 cm^2 + x^2 = s^2
6. Notice that we have a right triangle (the original one) with side lengths of 8 cm and x cm, and another right triangle (one of the smaller ones) created by the altitude, with side lengths of x cm and half the length of one side, s/2.
7. Since these two triangles are congruent, their corresponding sides are proportional. So we can set up the following equation:
x cm / (s/2) cm = (8 cm) / s cm
8. Cross-multiplying, we get:
x cm * s cm = (8 cm) * (s/2) cm
x * s = 4s
9. Dividing both sides by s, we find:
x = 4
Therefore, the length of one side of the equilateral triangle is 4 cm.
Finally, to calculate the perimeter of the triangle, we can multiply the length of one side by 3, since all sides of an equilateral triangle have the same length:
Perimeter = 4 cm * 3 = 12 cm
So, the perimeter of the equilateral triangle is 12 cm.