To solve for the length of the perpendicular bisector of a side in an equilateral triangle, we can follow these steps:
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Understand the Geometry: In an equilateral triangle, when we draw a perpendicular from a vertex to the opposite side, it serves as the height of the triangle. This line also bisects the side it meets (the base).
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Identify Key Values: The side length of the equilateral triangle is given as 36 inches.
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Relationship of Right Triangles in Equilateral Triangle: When the height (which also serves as the perpendicular bisector of the base) is drawn, it creates two 30-60-90 right triangles. In a 30-60-90 triangle:
- The sides opposite the 30°, 60°, and 90° angles have ratios of 1:√3:2 respectively.
- If the length of the side opposite the 60° angle (the height we want to find) is \( h \), and the hypotenuse is the side of the equilateral triangle (36 inches), then the side opposite the 30° angle (which is half the base, or \( 36/2 = 18 \) inches) corresponds to the shorter leg.
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Set Up the Equation: In our 30-60-90 triangle:
- If we take the shorter leg (half of the base) as 18 inches:
- The longer leg (the height \( h \)) is given by \( h = 18 \sqrt{3} \) inches (since it is opposite the 60° angle).
Therefore, the length of the perpendicular bisector (height) of the triangle is:
18√3 inches.
Thus, the correct answer is: 18√3 inches.