If the perimeter of an equilateral triangle is 24 inches, what is the length of the side that is opposite the 30-degree angle of the 30-60-90 triangle that is formed by half of the equilateral triangle?

Answer 1

To determine the length of the side opposite the 30-degree angle in the 30-60-90 triangle formed by halving an equilateral triangle with a perimeter of 24 inches, we start by finding the length of each side of the equilateral triangle.

1. Calculate the length of each side of the equilateral triangle: $$\text{Perimeter}=3\times \text{side length}$$ Given the perimeter is 24 inches: $$24=3\times \text{side length}$$ To find the side length: $$\text{side length}=\frac{24}{3}=8\text{ inches}$$

2. Find the altitude of the equilateral triangle: When you drop a perpendicular (altitude) from one vertex to the opposite side, this creates two 30-60-90 triangles. The altitude bisects the triangle and creates two smaller triangles. In a 30-60-90 triangle, the relationships between the sides are as follows:

   • The side opposite the 30-degree angle (half of the base of the equilateral triangle) is $x$.
   • The side opposite the 60-degree angle (the altitude) is $x\sqrt{3}$.
   • The hypotenuse (the side of the equilateral triangle) is $2x$.

3. Use the side length to find $x$: The hypotenuse of the 30-60-90 triangle is equal to the side length of the equilateral triangle, which is 8 inches: $$2x=8$$ To find $x$: $$x=\frac{8}{2}=4\text{ inches}$$

4. Identify the sides of the 30-60-90 triangle:

   • The side opposite the 30-degree angle is $4$ inches.
   • The side opposite the 60-degree angle (the altitude) is: $$x\sqrt{3}=4\sqrt{3}\text{ inches}$$

5. Conclusion: The length of the side opposite the 30-degree angle in the 30-60-90 triangle formed by halving the equilateral triangle is: $$\boxed{{\displaystyle 4\text{ inches}}}$$