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.
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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: \[ \text{side length} = \frac{24}{3} = 8 \text{ inches} \]
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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 \( 2x \).
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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 \( x \): \[ x = \frac{8}{2} = 4 \text{ inches} \]
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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: \[ x\sqrt{3} = 4\sqrt{3} \text{ inches} \]
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Conclusion: The length of the side opposite the 30-degree angle in the 30-60-90 triangle formed by halving the equilateral triangle is: \[ \boxed{4 \text{ inches}} \]
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