To find the length of the perpendicular bisector of a side of an equilateral triangle, we can first note the relevant properties of an equilateral triangle. In this case, we have an equilateral triangle with each side measuring 36 inches.
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Identify the altitude: The perpendicular bisector of a side in an equilateral triangle also serves as the altitude of the triangle. Therefore, we can calculate the altitude using the properties of a 30-60-90 triangle.
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Divide the triangle: When we draw the altitude from one vertex to the midpoint of the opposite side, we create two 30-60-90 right triangles. In this case, each right triangle will have:
- The hypotenuse equal to the side of the equilateral triangle (36 inches).
- The longer leg (the altitude) which we are trying to find.
- The shorter leg (half the length of the side opposite the vertex) which is \( \frac{36}{2} = 18 \) inches.
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Using the ratios: In a 30-60-90 triangle, the ratios of the sides are:
- Short leg (opposite the 30-degree angle): \( x \)
- Long leg (opposite the 60-degree angle): \( x\sqrt{3} \)
- Hypotenuse: \( 2x \)
Where \( x \) is half the side length. In our scenario:
- \( 2x = 36 \) (the hypotenuse).
- Therefore, \( x = 18 \).
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Calculate the altitude: The length of the altitude (the long leg) is: \[ \text{Altitude} = x\sqrt{3} = 18\sqrt{3} \text{ inches} \]
Thus, the length of the perpendicular bisector of any side of the equilateral triangle is \( 18\sqrt{3} \) inches, approximately \( 31.18 \) inches when you compute the decimal value of \( 18\sqrt{3} \).